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\section{Introduction}
This paper studies the stability of traveling wave solutions to scalar hyperbolic equations of the form
\begin{equation}
\label{hypAC}
\tau u_{tt} + g(u,\tau) u_t = u_{xx} + f(u),
\end{equation}
where $u$ is a scalar, $x \in \mathbb{R}$, $t > 0$, and $\tau \geq 0$ is a constant. Note that \eqref{hypAC} is a nonlinear wave equation with a ``damping term", $g$, and a nonlinear reaction term $f$. Hyperbolic equations of this form often support traveling wave solutions, also called traveling fronts, which are special solutions describing coherent structures which propagate along a particular direction with a certain wave speed. In a previous contribution \cite{LMPS16}, we analyzed the existence and stability of propagating fronts for a one-dimensional model which is a particular case of equation \eqref{hypAC}, called the \textit{Allen-Cahn equation with relaxation}. The motivation for the present study is to explore both the existence and the stability of such configurations for a wider class of equations, which arises in other contexts.
We make the following assumptions. First, the reaction function $f :
\mathbb{R} \to \mathbb{R}$ is supposed to be of \textit{bistable type}\footnote{also called of Nagumo \cite{NAY62,McKe70}, or
Allen-Cahn \cite{AlCa79} type.}, that is, $f \in C^2([0,1];\mathbb{R})$ has two stable
equilibria at $u=0, u=1$, and one unstable equilibrium point at $u = \alpha \in (0,1)$, more precisely,
\begin{equation}
\label{H1}
\tag{H1}
\begin{aligned}
&f(0)=f(\alpha)=f(1)=0,
&\qquad &f'(0), f'(1)<0,\quad f'(\alpha)>0,\\
&f(u)>0\textrm{ for all } \, u \in(\alpha,1),
&\qquad &f(u)<0\textrm{ for all } \, u \in (0,\alpha),
\end{aligned}
\end{equation}
for a certain $\alpha \in (0,1)$. A well-known example is the widely used cubic polynomial
\begin{equation}
\label{cubicf}
f(u)= u(1-u)(u-\alpha),
\end{equation}
with $\alpha \in (0,1)$.
Reaction functions of bistable type arise in many models of natural phenomena, such as kinetics of
biomolecular reactions (cf. Mikha{\u\i}lov \cite{Mik94}), nerve conduction (see, e.g.,
Lieberstein \cite{Lbr67a}, McKean \cite{McKe70}) and electrothermal instability (cf. Iz\'us \textit{et al.}
\cite{IDRWZB95}). In terms of continuous descriptions of the spread of biological populations, it is
often applied to kinetics exhibiting positive growth rate for population densities over a threshold
value ($u > \alpha$), and decay for densities below such value ($u < \alpha$). The latter is often
described as the \textit{Allee effect}, in which aggregation can improve the survival rate of
individuals (see Murray \cite{MurI3ed}).
Secondly, we are going to assume that the damping coefficient $g = g(u,\tau)$ in equation
\eqref{hypAC} is regular enough and strictly positive. More precisely, we suppose that for some fixed value
$\tau_m>0$, there holds
\begin{equation}
\label{H2}
\tag{H2}
g \in C^1(\mathbb{R} \times [0, \tau_m]), \;\; \; \text{and,} \quad \inf\left\{g(u, \tau) : u\in\mathbb{R},
\tau\in (0,\tau_m)\right\}\geq \delta_0>0,
\end{equation}
for some $\delta_0 > 0$ independent of $\tau_m$.
Assumption \eqref{H2} is an extension of the previously studied case of the Allen-Cahn model with relaxation
\cite{LMPS16}, where
\begin{equation}
\label{ACrelax}
g(u,\tau) = 1 - \tau f'(u),
\end{equation}
and with $\tau > 0$ bounded above by the characteristic relaxation time associated to the reaction,
\[
0 \leq \tau < \tau_m := \frac{1}{\max_{u \in [0,1]} |f'(u)|}.
\]
for which, clearly, $g(u,\tau) > 0$. If $F$ is an antiderivative such that $F'=-f$ with $F(0)=0$, that is,
\begin{equation*}
F(u):=-\int_{0}^{u} f(v)\,dv,
\end{equation*}
then $F$ can be interpreted as the Allen-Cahn two-well potential (see Figure \ref{figpotAC}).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=5.75cm]{bistable}
\includegraphics[width=5.75cm]{twowells}
\end{center}
\caption{\footnotesize The bistable cubic function $f(u)=u(1-u)(u-0.4)$ (left)
and the corresponding two-well potential $F$ (right).\label{figpotAC}}
\end{figure}
Another example of interest is the \textit{nonlinear telegrapher's
equation} \cite{Holm1}, where
\begin{equation}
\label{telegr}
g(u,\tau) \equiv 1,
\end{equation}
for all $u \in \mathbb{R}$ and $\tau \geq 0$.
\begin{remark}
There exist situations where the appearance of a diffusion coefficient $\varepsilon > 0$ in \eqref{hypAC},
\[
\tau u_{tt} + g(u,\tau) u_t = \varepsilon u_{xx} + f(u),
\]
is important, for example, in the study of slow motion of solutions or their \textit{metastability}
\cite{FLM17,Fol17}, when $0 < \varepsilon \ll 1$ is supposed to be small. For the problem of existence and stability of fronts, however,
the size of
$\varepsilon$ plays no role, and by rescaling the space variable, $x\mapsto x/\varepsilon$, we recover
equation
\eqref{hypAC}. Therefore, our analysis also applies to the more general model with arbitrary (constant)
diffusion and we can work with equation \eqref{hypAC} directly without loss of generality.
\end{remark}
In this paper we establish the spectral stability of traveling fronts for \eqref{hypAC} under the sole structural assumptions \eqref{H1} and \eqref{H2}, which include many models in population dynamics, microstructures and relaxation mechanisms, among others. In Section \ref{secexist} we prove that traveling fronts exist and provide some of their more important features and properties. Section \ref{secperturb} contains the perturbation problem and describes how to formulate a natural spectral problem (after linearization of the equation around the front), whose analysis encodes the most fundamental stability properties. We show that there exists two different but equivalent ways to formulate the spectral problem. In Section \ref{secess} we analyze the asymptotic systems associated to the perturbed equations and locate the essential spectrum. Section \ref{secptsp} contains the proof that the point spectrum is stable (via energy estimates in the frequency regime), the simplicity of the eigenvalue zero associated to translation, as well as the statement of our main result (see Theorem \ref{mainthm}). Finally, in section \ref{secdisc} we make some concluding remarks.
\section{Structure of traveling fronts}
\label{secexist}
In this section we review the existence theory and structural properties of front solutions to equations of
the form \eqref{hypAC}. In a recent contribution, Gilding and Kersner \cite{GiKe15} established the necessary
and sufficient conditions for the existence of traveling wave solutions to equation \eqref{hypAC} with
reaction function of bistable type under the assumption of positive damping $g > 0$. The authors make use of an
integral equation approach. For completeness, in this section we present an existence result which applies a
different technique based on the computation of the index of a rotating vector field of the dynamical system
with respect to the velocity (in the sense of Perko \cite{Per1}); this proof resembles our previous analysis
in the particular case of the relaxed Allen-Cahn model \cite{LMPS16}. With this approach we are able to
derive further structural properties, such as the exponential decay of the solutions and a variational formula for the
(unique) wave speed, which are not available from the integral formulation in \cite{GiKe15}.
\subsection{Existence}
\label{sect:existence}
We look for solutions to \eqref{hypAC} of the form
\begin{equation*}
u(x,t)=U(\xi)\quad\textrm{with}\quad\xi=x-ct,
\qquad\textrm{and}\qquad
U(-\infty)=0,\quad
U(+\infty)=1.
\end{equation*}
Substituting into \eqref{hypAC}, we obtain the equation
\begin{equation}
\label{twode0}
(1-c^2\tau)U''+c\,g(U,\tau)U'+f(U)=0,
\end{equation}
where $' \, := d/d\xi$.
\begin{proposition}\label{prop:properties}
Let assumptions \eqref{H1} and \eqref{H2} be satisfied,
and let $U=U(\xi)$ be a solution to \eqref{twode0} together with the asymptotic
conditions $U(-\infty)=0$ and $U(+\infty)=1$. Then,
\par
\textit{(i)} (speed sign) the velocity $c$ has the same sign of $-\int_{0}^{1} f(u)\,du$;\par
\textit{(ii)} (subcharacteristic condition) the velocity $c$ necessarily satisfies
\begin{equation}
\label{subchar}
c^2\tau < 1.
\end{equation}
\end{proposition}
\begin{proof}
\smartqed
(i) Multiplying equation \eqref{twode0} by $U'$ and integrating in $\mathbb{R}$, we obtain
\begin{equation*}
c\int_{\mathbb{R}} g(U,\tau)\left|U'\right|^2\,dx=F(1)-F(0).
\end{equation*}
where $F'=-f$. Thus, $\mathrm{sgn}(c) = \mathrm{sgn}(F(1) - F(0))$, as $g(U,\tau) > 0$.
(ii) The case $c = 0$ is manifest. If $c > 0$ then multiply equation \eqref{twode0} by $U'$. This yields,
\[
(1-c^2 \tau)U'' U' + cg(U,\tau)|U'|^2 + f(U) U' = 0.
\]
Since $f = - F'$ last equation is equivalent to
\begin{equation}
\label{eqfive}
\Big( \tfrac{1}{2}(1-c^2 \tau) |U'|^2 - F(U) \Big)' + cg(U,\tau) |U'|^2 = 0.
\end{equation}
Integrate equation \eqref{eqfive} in $(\xi,+\infty)$, to
obtain
\begin{equation}
\label{launo}
\tfrac{1}{2}(1-c^2 \tau) |U'(\xi)|^2 = F(U(\xi)) - F(1) + c \int_{\xi}^{+\infty} g(U(s),\tau)|U'(s)|^2
\, ds,
\end{equation}
and choose $\xi \gg 1$, large enough so that $U(\xi) \in (\alpha, 1)$ (as $U(+\infty) = 1$).
Since $f(u) > 0$ for $u \in (\alpha, 1)$ and $U(\xi) \in (\alpha, 1)$, clearly
\[
F(U(\xi)) - F(1) = \int_{U(\xi)}^1 f(s) \, ds > 0.
\]
Since we are assuming $c > 0$ and since $g(U,\tau) > 0$, clearly the right hand side of \eqref{launo} is positive, yielding $1 > c^2
\tau$. The case $c<0$ can be treated similarly.
\qed \end{proof}
\begin{remark}
Notice that if $F(0)=F(1)$, then the speed $c$ is necessarily zero and the equation for the profile reduces
to the one for traveling waves for the parabolic Allen-Cahn equation.
\end{remark}
We now prove an auxiliary result.
\begin{proposition}\label{prop:auxode}
Let assumptions \eqref{H1} - \eqref{H2} be satisfied.
Then there exists a unique value $\gamma\in\mathbb{R}$, denoted by $\gamma_\ast=\gamma_\ast(\tau)$,
such that the equation
\begin{equation}\label{auxode}
V''+\gamma\,g(V,\tau)V'+f(V)=0
\end{equation}
has a monotone increasing solution, $V=V(\xi)$ with asymptotic limits $V(-\infty)=0$ and $V(+\infty)=1$.
\end{proposition}
The proof of Proposition \ref{prop:auxode} consists of showing that there exists a heteroclinic
connection between the singular points $(V,V')=(0,0)$ and $(V,V')=(1,0)$.
We follow a standard shooting argument starting from the local analysis near the asymptotic
states, and use the special dependence with respect to the parameter $\gamma$ to
show that there is a single value $\gamma_\ast$ for which there exists a connecting orbit.
The strategy closely resembles the one presented in H\"arterich and Mascia \cite{HaeMa1}.
For shortness, we drop the dependence of $g$ with respect to $\tau$.
\begin{proof}[of Proposition \ref{prop:auxode}]
\smartqed
The second order differential equation \eqref{auxode} can be rewritten as
\begin{equation}\label{firstorder}
\left\{\begin{aligned}
V'&=\Phi(V,W;\gamma):=W,\\
W'&=\Psi(V,W;\gamma):=-f(V)-\gamma\,g(V)\,W,
\end{aligned}\right.
\end{equation}
possessing the two singular points $(0,0)$ and $(1,0)$.
1. Linearizing at $(\bar u,0)$, we obtain the matrix
\begin{equation*}
\begin{pmatrix}
\partial_V \Phi & &\partial_W \Phi\\
\partial_V \Psi & &\partial_W \Psi
\end{pmatrix}
=\begin{pmatrix}
0 & &1\\
-f'(\bar u) & & \, -\gamma\,g(\bar u)
\end{pmatrix}.
\end{equation*}
In particular, since $f'(0)$ and $f'(1)$ are negative, $(0,0)$ and $(1,0)$
are saddles for \eqref{firstorder}.
The positive eigenvalue $\mu^+_0$ at $(0,0)$ and the negative eigenvalue
$\mu^-_1$ at $(1,0)$ are
\begin{equation*}
\begin{aligned}
\mu^+_0&=\frac12\left(\sqrt{(\gamma\,g(0))^2-4f'(0)}-\gamma\,g(0)\right),
\\
\mu^-_1&=-\frac12\left(\sqrt{(\gamma\,g(1))^2-4f'(1)}+\gamma\,g(1)\right).
\end{aligned}
\end{equation*}
We denote by $\mathcal{U}_0(\gamma)$ the intersection of the unstable manifold of $(0,0)$
and the set $\{(V,W)\,:\,W>0\}$, and by $\mathcal{S}_1(\gamma)$ the intersection of the
unstable manifold of $(1,0)$ and the set $\{(V,W)\,:\,W>0\}$.
2. Let $\gamma<0$ and $\hat W>M/c_0|\gamma|$, where $M:=\max\{f(u)\,:\,u\in(\alpha,1)\}$.
The solution trajectory passing through $(\alpha,W_0)$ is the graph of the solution
$\omega=\omega(V)$ to the Cauchy problem
\begin{equation}\label{trajeq}
\frac{d\omega}{dV}=-\frac{f(V)}{\omega}-\gamma\,g(V),
\end{equation}
with initial condition $\omega(\alpha)=\hat W$.
Denote its interval of maximal existence by $I$, and observe that $\omega$ is strictly increasing in $I\cap
[\alpha,1]$.
Indeed, since $d\omega/dV(\alpha)=-\gamma\,g(\alpha)>0$, the function $\omega$ is strictly
increasing for $V\in(\alpha,\alpha+\delta)$ for some $\delta>0$.
Moreover, if $\omega>\hat W$ and $V\in[\alpha,1]$ there holds
\begin{equation*}
\frac{d\omega}{dV}\geq -\frac{M}{\hat W}+c_0|\gamma|>0,
\end{equation*}
and the claim follows from a standard continuation argument.
As a consequence, the derivative of $\omega$ is a priori bounded and
the interval $I$ contains the interval $[\alpha,1]$.
Since the vector field $(\Phi,\Psi)$ point downward along the segment $(\alpha,1)\times\{0\}$,
the curve $\mathcal{S}_1(\gamma)$ intersect the line $V=\alpha$ at some value $W_1(\gamma)\geq 0$
for $\gamma<0$.
Similar arguments show that $\mathcal{U}_0(\gamma)$ intersects the line
$V=\alpha$ at some $W_0(\gamma)$ for $\gamma>0$.
3. Since
\begin{equation*}
\det\begin{pmatrix}
\Phi & &\Psi\\
\partial_\gamma\Phi & & \partial_\gamma \Psi
\end{pmatrix}
=\det\begin{pmatrix}
W & & \, -f-\gamma\,g\,W\\
0 & & \, -g\,W
\end{pmatrix}=-g\,W^2\leq -c_0 W^2\leq 0,
\end{equation*}
the vector field defining the differential system is a \textit{rotated vector field}
with respect to the parameter $\gamma$ (see Perko \cite{Per1}).
As a consequence, the graphs $\mathcal{U}_0(\gamma)$ and $\mathcal{S}_1(\gamma)$
rotate clockwise as the parameter $\gamma$ increases.
Therefore, the map $W_{0}=W_{0}(\gamma)$ is monotone decreasing in $(0,+\infty)$
and the map $W_{1}(\gamma)=W_{1}(\gamma)$ is monotone increasing in $(-\infty,0)$.
4. If $\bar V$ is a relative maximum point for a solution $\omega$ to \eqref{trajeq},
then
\begin{equation*}
|\omega(\bar V)|=\frac{|f(\bar V)|}{|\gamma|\,g(\bar V)}\leq \frac{M}{c_0|\gamma|},
\end{equation*}
where $M$ is the maximum of $|f|$ in $(0,1)$.
Thus, $W_0(\gamma)\to 0$ as $\gamma\to+\infty$
and $W_1(\gamma)\to 0$ as $\gamma\to-\infty$.
Following Hadeler \cite{Had1}, let us note that one can also prove that there exist values
$\gamma_\pm$ with $\gamma_0<0<\gamma_1$ such that $W_0(\gamma_0)=0$ and
$W_1(\gamma_1)=0$.
Then, for any $\gamma\geq \gamma_0$ the trajectory $\mathcal{U}_0(\gamma)$
describes a heteroclinic connection between $0$ and $\alpha$;
similarly, for any $\gamma\leq \gamma_1$ the trajectory $\mathcal{S}_1(\gamma)$
describes a heteroclinic connection between $\alpha$ and $1$.
5. From monotonicity of $W_{0}$ and $W_{1}$, we infer that they both have limits as $\gamma\to 0$.
Additionally, the trajectory equation \eqref{trajeq} shows that such limiting values $W_0(0)$ and
$W_{1}(0)$ are finite and can be computed explictly, taking advantage of the conserved quantity
$W^2-2F(V)$, yielding
\begin{equation*}
W_0(0)=\sqrt{2\bigl(F(\alpha)-F(0)\bigr)}
\qquad\textrm{and}\qquad
W_1(0)=\sqrt{2\bigl(F(\alpha)-F(1)\bigr)}.
\end{equation*}
Since the solution depends continuously with respect to the parameter $\gamma$,
there exist $\gamma_0, \gamma_1$ with $-\infty\leq \gamma_0<0<\gamma_1\leq +\infty$,
such that $W_0$ is defined (and monotone decreasing) in $(\gamma_0,+\infty)$ and $W_1$
is defined (and monotone increasing) in $(-\infty,\gamma_1)$.
If $\gamma_0$ is finite, $W_0\to+\infty$ as $\gamma\to\gamma_0^+$;
similarly, if $\gamma_1$ is finite, $W_1\to+\infty$ as $\gamma\to\gamma_1^-$.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=9cm, height=6.5cm]{GraphWsxWdx1}
\end{center}
\caption{\footnotesize Graphs of the curves $\mathcal{U}_0(\gamma)$ (left) and $\mathcal{S}_1(\gamma)$ (right)
in the plane $(V,W)$ for different values of $\gamma$:
dashed line $\gamma=0$; dotted line $\gamma=-0.40$; and continuous line $\gamma=-0.32$. Here we considered the case of a cubic nonlinearity $f(u)=u(1-u)(u-\alpha)$ with $\alpha=0.3$, and damping term of Cattaneo-Maxwell type, $g(u;\tau)=1-\tau f'(u)$, where $\tau=1$.
}
\end{figure}
6. Let us consider the difference function $h:=W_1-W_0$ defined in $(\gamma_0,\gamma_1)$.
As a consequence of the properties of $W_0$ and $W_1$, we infer that $h$ is continuous,
monotone increasing and such that
\begin{equation*}
\liminf_{\gamma\to\gamma_0^+} h(\gamma)<0,\quad
\liminf_{\gamma\to\gamma_1^-} h(\gamma)>0.
\end{equation*}
In particular, there exists a unique $\gamma_\ast$ such that
$W_0(\gamma_\ast)=W_1(\gamma_\ast)$.
For such critical value, the conjuction of the curves $\mathcal{U}_0(\gamma_\ast)$ and
$\mathcal{S}_1(\gamma_\ast)$ gives the desired connection.
Uniqueness of the wave speed $\gamma_\ast$ follows from the monotonicity
of the functions $W_0$ and $W_1$.
\qed \end{proof}
\begin{remark}
Equation \eqref{auxode} arises also in the case of reaction-diffusion equations
with density-dependent diffusion
\begin{equation*}
w_t = \varphi(w)_{xx} + f(w),
\end{equation*}
where $\varphi$ is a strictly increasing function. Inserting the traveling wave profile \textit{ansatz}
$w(x,t)=W(x-\gamma t)$ and setting $V:=\phi(W)$ yields
\begin{equation*}
\frac{d^2V}{d\eta^2}+\gamma \psi'(V)\frac{dV}{d\eta}+f(\psi(V))=0,
\end{equation*}
where $\psi$ is the inverse function of $\phi$.
In fact, existence of heteroclinic solutions for \eqref{auxode} could be also proved by
appropriately changing the dependent variable $V$
and applying the general result proved by Engler \cite{Eng2} that relates the existence
of traveling wave solutions of reaction-diffusion equations with constant diffusion
coefficient to the ones of the density-dependent diffusion coefficient case.
\end{remark}
\begin{example}
\label{exAllenCahn}
In the special case of a nonlinear telegrapher's equation with cubic reaction function, namely,
\begin{equation}\label{fbist_g1}
g(u,\tau)=1,\qquad f(u)=\kappa\,u(1-u)(u-\alpha),
\end{equation}
we can look for $W=V'$ with the form $W(V) = AV(1-V)$, where $A$ is a constant to be determined.
Inserting in \eqref{auxode}, we deduce the following constraints on $A$ and $\gamma$
\begin{equation*}
A^2+\gamma\,A-\kappa\,\alpha=0,\qquad 2A^2-\kappa=0,
\end{equation*}
giving the explicit formulas $A=\sqrt{\kappa/2}$ and
\begin{equation}\label{gast_fbist_g1}
\gamma_\ast=\gamma_{{}_{\textrm{\tiny AC}}}:=\sqrt{\frac{2}{\kappa}}\left(\alpha-\frac12\right),
\end{equation}
which is the speed of propagation for the (parabolic) Allen--Cahn equation.
In the significant relaxation case $g(u,\tau)=1-\tau f'(u)$, the same simplification does not hold and
an analogous explicit formula for the critical speed $\gamma_\ast$ is not available. However, as in the case
of the standard Allen--Cahn equation, it is possible to establish
a min-max variational characterization for the critical speed $\gamma_\ast$ (cf. Hamel \cite{Hame99}; see also
\cite{MFF1}).
\end{example}
\begin{proposition}[variational formula for the speed]
\label{prop:minmax}
Let assumptions \eqref{H1} - \eqref{H2} be satisfied.
Set
\begin{equation*}
\mathcal{W}:=\{W\in C^2(\mathbb{R})\,:\, W(x)\in(0,1),\, W'(x)>0\;\,\text{for any}\, \;x\in\mathbb{R}\}.
\end{equation*}
Then the speed $\gamma_\ast$ defined in Proposition \ref{prop:auxode} is such that
\begin{equation}\label{minmax}
\gamma_\ast=-\inf_{W\in \mathcal W} \sup_{x\in\mathbb{R}}\frac{W''+f(W)}{g(W)\,W'}
=-\sup_{W\in \mathcal W} \inf_{x\in\mathbb{R}}\frac{W''+f(W)}{g(W)\,W'}.
\end{equation}
\end{proposition}
\begin{proof}
\smartqed
We give a sketch of the proof.
Denote by $V$ the traveling profile given by Proposition \ref{prop:auxode};
then there holds $\gamma_\ast=-(V''+f(V))/g(V)V'$.
Since $V\in\mathcal W$, we infer the inequalities
\begin{equation*}
\underline{\gamma}:= \inf_{W\in \mathcal W} \sup_{x\in\mathbb{R}}\frac{-\bigl(W''+f(W)\bigr)}{g(W)\,W'}
\leq \gamma_\ast
\leq \overline{\gamma}:=\sup_{W\in \mathcal W} \inf_{x\in\mathbb{R}}\frac{-\bigl(W''+f(W)\bigr)}{g(W)\,W'}.
\end{equation*}
If $\gamma_\ast<\overline{\gamma}$ then for any $\gamma\in(\gamma_\ast,\overline{\gamma})$,
there exists a function $W\in\mathcal W$ such that
\begin{equation*}
\inf_{x\in\mathbb{R}}\frac{-\bigl(W''+f(W)\bigr)}{g(W)\,W'}\geq \gamma.
\end{equation*}
As a consequence, we deduce
\begin{equation*}
W''+\gamma\,g(W)\,W'+f(W)\leq 0\leq
(\gamma-\gamma_\ast) g(V)\,V'=V''+\gamma\,g(V)\,V'+f(V),
\end{equation*}
showing that $W$ and $U$ are, respectively, super- and subsolution for
\begin{equation}\label{elliptic}
U''+\gamma\,g(U)\,U'+f(U)=0.
\end{equation}
Invoking a monotonicity argument \cite{ProWe84}, we deduce the existence of a solution
$U$ to \eqref{elliptic}
such that $V\leq U\leq W$,
thus satisfying, in particular, the asymptotic conditions $U(-\infty)=0$ and
$U(+\infty)=1$.
Such statement contradicts the uniqueness of the speed $\gamma_\ast$
given in Proposition \ref{prop:auxode}.
Thus, $\gamma_\ast=\overline{\gamma}$. Proving in an analogous manner the equality
$\gamma_\ast=\underline{\gamma}$,
we deduce formula \eqref{minmax}.
\qed \end{proof}
Independently from the variational characterization of the wave speed, the existence of a solution
for \eqref{twode0} with appropriate asymptotic values is a straightforward
consequence of Proposition \ref{prop:auxode}.
The relation between the speed $\gamma$ of Proposition \ref{prop:auxode} and $c$ for
\eqref{twode0} guarantees the uniqueness of the speed for the hyperbolic
Allen--Cahn equation.
\begin{theorem}[existence of a traveling front]
\label{theoexists}
Under assumptions \eqref{H1} - \eqref{H2} there exists a unique value $c\in\mathbb{R}$, denoted by
$c_\ast=c_\ast(\tau)$,
such that the equation
\begin{equation}\label{twode}
(1-c^2\tau)U''+c\,g(U,\tau)U'+f(U)=0.
\end{equation}
has a monotone increasing front solution $U=U(\xi)$ with $U(-\infty)=0$ and $U(+\infty)=1$. The value
$c_\ast=c_\ast(\tau)$ is related to $\gamma_\ast=\gamma_\ast(\tau)$
of Proposition \ref{prop:auxode} by the relation
\begin{equation}\label{castgast}
c_\ast=\frac{\gamma_\ast}{\sqrt{1+\tau\,\gamma_\ast^2}}.
\end{equation}
\end{theorem}
\begin{proof}
\smartqed
Thanks to the subcharacteristic condition \eqref{subchar}, we can
restrict our attention to
$c\in(-1/\sqrt{\tau},1/\sqrt{\tau})$.
By applying the change of variables
\begin{equation*}
\sqrt{1-c^2\tau}\frac{d}{d\xi}=\frac{d}{d\eta},
\end{equation*}
and setting $\gamma=\gamma(c)=c/\sqrt{1-c^2\tau}$,
equation \eqref{twode} transforms into \eqref{auxode}.
Then the profile existence and uniqueness statement follows since
$\gamma=\gamma(c)$ is increasing and $\gamma(\pm 1 /\sqrt{\tau})=\pm\infty$.
Relation \eqref{castgast} is obtained by inverting the function $\gamma = \gamma(c)$.
\qed \end{proof}
\subsection{Exponential decay}\label{secexpdecay}
As a consequence of the analysis in Proposition \ref{prop:auxode} and Theorem \ref{theoexists}, the profile function decays to its asymptotic limits exponentially fast.
\begin{lemma}[exponential decay of the profile]
\label{lemexpdecay}
For each $\tau \geq 0$ the front solution and its derivatives satisfy
\begin{equation}
\label{expdecay}
|\partial_\xi^j(U(\xi) - U_\pm)| \leq C e^{-\eta|\xi|},
\end{equation}
for all $\xi \in \mathbb{R}$, $j = 0,1,2$, with uniform constants $C > 0$ and $\eta > 0$.
\end{lemma}
\begin{proof}
\smartqed
Suppose that $U = U(\xi)$ is the profile function of Theorem \ref{theoexists}, traveling with speed $c = c_*(\tau)$. As before, $\xi = x - ct$ and $\, ' = d/d\xi$. If we denote $V = U'$ then $(U,V) = (U,V)(\xi)$ is an heteroclinic connection between the rest points
\[
(U_+, V_+) = (1,0) \quad \text{and,} \quad (U_-,V_-) = (0,0),
\]
as $\xi \to \pm \infty$, of the first order system
\begin{equation}
\label{firstODE}
\begin{pmatrix}U \\ V
\end{pmatrix}' = \begin{pmatrix} V \\ - (1-c^2 \tau)^{-1} (f(U) +c g(U,\tau)V)
\end{pmatrix} =: \begin{pmatrix} \hat{\Phi} \\ \hat{\Psi}\end{pmatrix} (U,V).
\end{equation}
Linearizing around the asymptotic rest states we obtain
\[
\frac{D(\hat{\Phi}, \hat{\Psi})}{D(U,V)}(U_\pm, V_\pm) = \begin{pmatrix} 0 & & 1 \\ (1-c^2\tau)^{-1} |a_\pm| & & \, -(1-c^2\tau)^{-1}c b_\pm \end{pmatrix},
\]
where, in view of assumptions \eqref{H1} and \eqref{H2}, we have denoted $a_\pm = f'(U_\pm) < 0$ and $b_\pm = g(U_\pm,\tau) > 0$. Its eigenvalues are
\[
\mu_{1,2}^\pm = - \tfrac{1}{2} cb_\pm (1-c^2 \tau)^{-1} \pm \tfrac{1}{2} \sqrt{c^2 b_{\pm}^2(1-c^2
\tau)^{-2} + 4(1-c^2 \tau)^{-1}|a_\pm|},
\]
which are real and the asymptotic states are non-degenerate hyperbolic points. The positive eigenvalue at $(U_-,V_-) = (0,0)$ is
\[
\mu_2^- = - \tfrac{1}{2}cb_-(1-c^2 \tau)^{-1} + \tfrac{1}{2} \sqrt{c^2 b_{-}^2(1-c^2 \tau)^{-2} +
4(1-c^2 \tau)^{-1}|a_-|},
\]
and the orbit decays to $(U_-,V_-) = (0,0)$ with exponental rate $|(U,V)(\xi)| \leq C e^{\mu_2^- \xi}$ as
$\xi \to -\infty$ for some uniform $C > 0$. The negative eigenvalue at $(U_+,V_+) = (1,0)$ is
\[
\mu_1^+ = - \tfrac{1}{2}cb_+(1-c^2 \tau)^{-1} - \tfrac{1}{2} \sqrt{c^2 b_{+}^2(1-c^2 \tau)^{-2} +
4(1-c^2 \tau)^{-1}|a_+|},
\]
and the orbit decays as $|(U,V)(\xi) - (1,0)| \leq C e^{-|\mu_1^+|\xi}$, when $\xi \to +\infty$. Thus, if we define $\eta = \min \{\mu_2^-, |\mu_1^+|\} > 0$ we obtain the result. Notice that $\eta = \eta(\tau) > 0$ for each fixed $\tau \geq 0$ and that $V' = U''$ also decays exponentially fast.
\qed
\end{proof}
\section{Perturbation equations and the stability problem}
\label{secperturb}
In this section we derive the equation for a perturbation of the traveling front, linearize it around the
wave, and set up the associated spectral
problem.
For fixed $\tau > 0$ let $c = c_*(\tau) \in (-1/\sqrt{\tau},1/\sqrt{\tau})$ be the unique wave speed of the
traveling front of Theorem \ref{theoexists}. We then recast equation \eqref{hypAC} in the moving coordinate
frame and, with a slight abuse of notation, make the transformation $x \to x-ct$ so that the model equation
\eqref{hypAC} now reads
\begin{equation}
\label{newhypAC}
\tau u_{tt} - 2c\tau u_{xt} + g(u,\tau) u_t = (1-c^2 \tau) u_{xx} + c g(u,\tau) u_x + f(u).
\end{equation}
From this point on and for the rest of the paper $x$ will denote the (Galilean) moving variable and
the front profile $U = U(x)$ is now a stationary solution to \eqref{newhypAC}, satisfying
\begin{equation}
\label{nprofileq}
(1-c^2\tau) U_{xx} + cg(U,\tau) U_x + f(U) = 0.
\end{equation}
As before, the asymptotic limits are $U_+ = U(+\infty) = 1$ and $U_- = U(-\infty) = 0$. In view of Lemma
\ref{lemexpdecay} the convergence of $U$ to its asymptotic limits is exponential,
\begin{equation}
\label{expdec}
|\partial_x^j(U - U_\pm) (x)| \leq C e^{- \eta |x|},
\end{equation}
as $x \to \pm \infty$ and for some $C, \eta > 0$.
\begin{remark}
\label{remUreg}
By regularity of the profile and its exponential decay, it is clear that $U_x \in H^1(\mathbb{R})$.
Apply a bootstrapping argument to verify that, in fact, $U_x \in H^3(\mathbb{R})$. Details are left to the reader.
\end{remark}
\subsection{Equations for the perturbation and the spectral problem}
Let us consider solutions to \eqref{newhypAC} of the form $u(x,t) + U(x)$, where now $u = u(x,t)$ stands for
a perturbation of the front. Upon substitution, we obtain the following nonlinear equation for the
perturbation,
\begin{equation}
\label{nlpert}
\begin{aligned}
\tau u_{tt} - 2c\tau u_{xt} &+ g(u+U,\tau) u_t = \\ &=(1-c^2 \tau) u_{xx} + (1-c^2 \tau) U_{xx} + c
g(u+U,\tau)(
u_x + U_x) + f(u).
\end{aligned}
\end{equation}
Expand the nonlinear terms in Taylor series around $U$ and use the profile equation \eqref{nprofileq} to
write equation \eqref{nlpert} as
\[
\begin{aligned}
\tau u_{tt} - 2c\tau u_{xt} + g(U,\tau) u_t &= (1-c^2 \tau) u_{xx} + c g(U,\tau) u_x + (c
g_u(U,\tau)U_x + f'(U))u +\\ & \, + O(|uu_t|) + O(|uu_x|) + O(|u|^2).
\end{aligned}
\]
Let us define
\[
a(x) := c g(U,\tau)_x + f'(U), \quad b(x) := g(U,\tau) \, > \, 0.
\]
Dropping the nonlinear terms we arrive at the following linearized equation for the perturbation
\begin{equation}
\label{linequ}
\tau u_{tt} - 2c\tau u_{xt} + b(x) u_t = (1-c^2 \tau) u_{xx} + c b(x) u_x + a(x) u.
\end{equation}
Let us specialize the linear problem to solutions of the form $u(x,t) = e^{\lambda t} v(x)$, where $\lambda
\in \mathbb{C}$ is the spectral parameter and $v$ belongs to an appropriate Banach space $X$. The result is the
following spectral equation for $v$,
\begin{equation}
\label{specprob}
\lambda^2 \tau v - 2c \lambda \tau v_x + \lambda b(x) v = (1-c^2 \tau) v_{xx} + cb(x) v_x + a(x) v,
\end{equation}
for some $v \in X$, $\lambda \in \mathbb{C}$.
In this analysis we choose the perturbation space to be $X = L^2(\mathbb{R};\mathbb{C})$, and the domain of solutions to
\eqref{specprob} to be $\mathcal{D} = H^2(\mathbb{R};\mathbb{C})$. In the sequel, $L^2$ and $H^m$, with $m > 0$, will denote the complex spaces $L^2(\mathbb{R};\mathbb{C})$ and $H^m(\mathbb{R};\mathbb{C})$, respectively, except where it is explicitly stated otherwise.
\begin{remark}
\label{rempencil}
Notice that the spectral equation \eqref{specprob} is quadratic in $\lambda$. Under the substitution $\lambda
= i \zeta$ equation \eqref{specprob} can be written in terms of a
\textit{quadratic operator pencil} $\tilde {\mathcal{A}}(\zeta)$ (cf. Markus \cite{Markus88}), given by
\[
\tilde {\mathcal{A}} (\zeta) = \tilde {\mathcal{A}}_0 + \zeta \tilde {\mathcal{A}}_1 + \zeta^2 \tilde {\mathcal{A}}_2,
\]
with
\[
\begin{aligned}
\tilde {\mathcal{A}}_0 &= (1-c^2 \tau) \frac{d^2}{dx^2} + cb(x) \frac{d}{dx} + a(x),\\
\tilde {\mathcal{A}}_1 &= i 2 \tau \frac{d}{dx} - ib(x),\\
\tilde {\mathcal{A}}_2 &= \tau.
\end{aligned}
\]
It is easy to see that \eqref{specprob} is equivalent to $\tilde {\mathcal{A}}(\zeta) v = 0$. The transformation
$v_1 = v$, $v_2 = \lambda v - cv_x$ defines an
appropriate Cartesian product of the base space which allows us to write equation \eqref{specprob} as a
genuine eigenvalue problem in the form
\begin{equation}
\label{defcLtau}
\lambda \begin{pmatrix}
v_1 \\ v_2
\end{pmatrix} = \begin{pmatrix}
c \partial_x & & 1 \\ \tau^{-1} (\partial_x^2 + a(x)) & & \, c \partial_x - \tau^{-1}b(x)
\end{pmatrix}\begin{pmatrix}v_1 \\ v_2 \end{pmatrix} =: {\mathcal{L}}^\tau \begin{pmatrix}v_1 \\
v_2\end{pmatrix}.
\end{equation}
The linear operator ${\mathcal{L}}^\tau$ (densely defined in $L^2 \times L^2$ with domain
${\mathcal{D}}({\mathcal{L}}^\tau) = H^2 \times H^1$ for $\tau > 0$) is often called the \textit{companion matrix} to the pencil $\tilde
{\mathcal{A}}$ (see \cite{BrJoK14,KoMi14,LaSu1} for further information).
\end{remark}
\subsection{Reformulation as a first order system}
According to custom in the literature of stability of nonlinear waves \cite{AGJ90,KaPro13}, we now recast the
spectral problem \eqref{specprob} as a first order system in the frequency regime of the form
\begin{equation}
\label{Wsystem}
W_x = \mathbb{A}^\tau(x,\lambda) W,
\end{equation}
where $\lambda \in \mathbb{C}$ is a parameter and $\tau > 0$ is fixed. Indeed, making
\[
W= \begin{pmatrix}
v \\ v_x
\end{pmatrix},
\]
and noticing that because of the subcharacteristic condition (see Proposition \ref{prop:properties}
(ii)) there holds $1 - c^2 \tau > 0$, we obtain a first order ODE system of the form \eqref{Wsystem}
with coefficient matrix given by
\begin{equation}
\label{coeffA}
\mathbb{A}^\tau(x,\lambda) = (1 - c^2 \tau)^{-1}\begin{pmatrix}
0 & & 1-c^2\tau \\ \tau \lambda^2 + \lambda b(x) - a(x) & & \, -c(b(x) + 2
\tau \lambda)
\end{pmatrix}.
\end{equation}
Since $U(x) \to U_\pm$ as $x \to \pm \infty$, with $U_- = 0$, $U_+ = 1$, let us denote
\[
\begin{aligned}
a_\pm &= \lim_{x \to \pm \infty} a(x) = \lim_{x \to \pm \infty} \big( f'(U) + g_u(U,\tau)U_x \big) =
f'(U_\pm)
< 0,\\
b_\pm &= \lim_{x \to \pm \infty} b(x) = \lim_{x \to \pm \infty} g(U,\tau) = g(U_\pm,\tau) > 0,
\end{aligned}
\]
because $U_x \to 0$, $f'(1)$, $f'(0) < 0$ and $g(U,\tau) > 0$, by hypotheses \eqref{H1} and \eqref{H2}. In
this fashion, we denote the asymptotic coefficient matrices as
\begin{equation}
\label{asympcoeff}
\begin{aligned}
\mathbb{A}^\tau_\pm (\lambda) &:= \lim_{x \to \pm \infty} \mathbb{A}^\tau(x,\lambda) \\&= (1 - c^2\tau)^{-1} \begin{pmatrix}
0 & & 1 - c^2 \tau \\
\tau \lambda^2 + \lambda b_\pm + |a_\pm| & & \, -c(b_\pm + 2 \tau \lambda)
\end{pmatrix},
\end{aligned}
\end{equation}
for each $\tau \geq 0$, $\lambda \in \mathbb{C}$.
It is convenient to define the spectra and
resolvent of the spectral problem \eqref{specprob} in terms of the first order systems \eqref{Wsystem}.
Consider the following family of linear,
closed, densely defined operators
\[
{\mathcal{T}}^{\tau}(\lambda) : \bar {\mathcal{D}} \to L^2 \times L^2,
\]
\[
{\mathcal{T}}^{\tau}(\lambda) := \partial_x - \mathbb{A}^{\tau}(x,\lambda),
\]
with domain $\bar {\mathcal{D}} = H^1 \times H^1$, indexed by $\tau \geq 0$ and parametrized by $\lambda \in \mathbb{C}$. With
a
slight abuse of notation we call $W \in H^1 \times H^1$ an \textit{eigenfunction} associated to the
eigenvalue
$\lambda \in \mathbb{C}$ provided $W$ is a bounded solution to the equation
\[
{\mathcal{T}}^{\tau}(\lambda) W = W_x - \mathbb{A}^{\tau}(x,\lambda) W = 0.
\]
\begin{definition}[resolvent and spectra]
\label{defsigmatwo}
For fixed $\tau \geq 0$ we define,
\[
\begin{aligned}
\rho &:= \{\lambda \in \mathbb{C} \, : \, {\mathcal{T}}^{\tau}(\lambda) \,\text{ is injective and onto, and }
{\mathcal{T}}^{\tau}(\lambda)^{-1} \, \text{is bounded} \, \},\\
\sigma_\mathrm{\tiny{pt}} &:= \{ \lambda \in \mathbb{C}\,: \; {\mathcal{T}}^{\tau}(\lambda) \,\text{ is Fredholm with index zero and has a} \\
& \qquad \qquad \qquad \text{non-trivial kernel} \},\\
\sigma_\mathrm{\tiny{ess}} &:= \{ \lambda \in \mathbb{C}\,: \; {\mathcal{T}}^{\tau}(\lambda) \,\text{ is either not Fredholm or has index
different } \\
& \qquad \qquad \qquad \text{from zero} \}.
\end{aligned}
\]
The spectrum $\sigma$ of problem \eqref{specprob} is defined as $\sigma = \sigma_\mathrm{\tiny{ess}} \cup
\sigma_\mathrm{\tiny{pt}}$. Since ${\mathcal{T}}^{\tau}(\lambda)$ is closed, we know that $\rho = \mathbb{C} \backslash \sigma$ (cf.
Kato \cite{Kat80}).
\end{definition}
\begin{remark}
This definition of spectrum is due to Weyl \cite{We10}, making $\sigma_\mathrm{\tiny{ess}}$ a large set but easy to compute,
whereas $\sigma_\mathrm{\tiny{pt}}$ is a discrete set of isolated eigenvalues with finite multiplicity (see Remark 2.2.4
in \cite{KaPro13}). We remind the reader that a closed operator $\mathcal{L}$ is said to be
Fredholm if its range $\mathcal{R(L)}$ is closed, and both its nullity, $\text{nul}\,\mathcal{L} = \dim \ker
\mathcal{L}$, and its deficiency, $\mathrm{def} \,\mathcal{L} = \mathrm{codim} \, \mathcal{R(L)}$, are
finite. In such a case the index of ${\mathcal{L}}$ is defined as
$\text{ind}\, \mathcal{L} = \text{nul}\, \mathcal{L} - \mathrm{def} \, \mathcal{L}$ (cf. \cite{Kat80}).
\end{remark}
For each $\tau \geq 0$ we can write the coefficients as
\[
\mathbb{A}^\tau(x,\lambda) = \mathbb{A}^\tau_0(x) + \lambda \mathbb{A}^\tau_1(x) + \lambda^2 \mathbb{A}^\tau_2(x),
\]
where
\[
\mathbb{A}^\tau_0(x) = (1-c^2\tau)^{-1}\begin{pmatrix}
0 & & 1 - c^2\tau \\ -a(x) & & -cb(x)
\end{pmatrix},
\]
\[
\mathbb{A}^\tau_1(x) = (1-c^2\tau)^{-1}\begin{pmatrix}
0 & & 0 \\ b(x) & & -2c\tau
\end{pmatrix},
\]
\[
\mathbb{A}^\tau_2(x) = (1-c^2\tau)^{-1}\begin{pmatrix}
0 & & 0 \\ \tau & & 0
\end{pmatrix}.
\]
Therefore, we may compute
\begin{equation}
\label{derivlamA}
\partial_\lambda \mathbb{A}^\tau(x,\lambda) = \mathbb{A}^\tau_1(x) + 2 \lambda \mathbb{A}^\tau_2(x).
\end{equation}
Furthermore, if we regard the coefficients \eqref{coeffA} as functions from $(\lambda,\tau)$ into $L^\infty$
then they are analytic in $\lambda$ (quadratic polynomial) and continuous in $\tau$.
We also define the algebraic and geometric multiplicities of the elements in the point spectrum as
follows.
\begin{definition}
\label{defmult}
Assume $\lambda \in \sigma_\mathrm{\tiny{pt}}$. Its geometric multiplicity (\textit{g.m.}) is the maximal number of linearly
independent elements in $\ker {\mathcal{T}}^{\tau}(\lambda)$. Suppose $\lambda \in \sigma_\mathrm{\tiny{pt}}$ has $g.m. = 1$, so that
$\ker {\mathcal{T}}^{\tau}(\lambda) =$ span $\{W_0\}$. We say $\lambda$ has algebraic multiplicity (\textit{a.m.})
equal to $m$ if we can solve
\[
{\mathcal{T}}^{\tau}(\lambda) W_j = \partial_\lambda \mathbb{A}^\tau(x,\lambda) W_{j-1},
\]
for each $j = 1, \ldots, m-1$, with $W_j \in H^1$, but there is no bounded $H^1$ solution $W$ to
\[
{\mathcal{T}}^{\tau}(\lambda) W = \partial_\lambda \mathbb{A}^\tau(x,\lambda) W_{m-1}.
\]
For an arbitrary eigenvalue $\lambda \in \sigma_\mathrm{\tiny{pt}}$ with $g.m.= l$, the algebraic multiplicity is defined as the
sum of the multiplicities $\sum_k^l m_k$ of a maximal set of linearly independent elements in $\ker
{\mathcal{T}}^{\tau}(\lambda) = $ span $\{W_1, \ldots, W_l\}$.
\end{definition}
\begin{remark}
Notice that, unlike the operator defined in \eqref{defcLtau}, the spectral problem formulated as a first order system is well defined also for $\tau = 0$, as
\begin{equation}
\label{coeffA0}
\mathbb{A}^0(x,\lambda) = \begin{pmatrix}
0 & & 1 \\ \lambda b(x) - a(x) & & \, -c b(x)
\end{pmatrix},
\end{equation}
where the coefficients $a(x) = f'(U) + g(U,0)_x$, $b(x) = g(U,0)$ and the speed $c = c(0)$ are evaluated at
$\tau = 0$.
\end{remark}
Finally we remark that, due to translation invariance, $\lambda = 0$ belongs to the point spectrum.
\begin{lemma}
\label{lemzeroeigenv}
For each $\tau \geq 0$, $0 \in \sigma_\mathrm{\tiny{pt}}$, with associated eigenfunction $\Phi = (U_x, U_{xx})^\top \in H^1
\times H^1$.
\end{lemma}
\begin{proof}
\smartqed
Follows by a direct calculation using the profile equation \eqref{nprofileq}.
Notice that $U_{x} \in H^2$ (see Remark \ref{remUreg}), so that $\Phi = (U_x, U_{xx})^\top \in \ker
{\mathcal{T}}^\tau(0) \subset H^1 \times H^1$ is indeed an eigenfunction.
\qed \end{proof}
\subsection{Spectral equivalence}
\label{secequiv}
The seasoned reader might rightfully ask what is the relation between the spectrum of Definition \ref{defsigmatwo}, and the standard spectrum of the family of operators ${\mathcal{L}}^\tau$ defined in \eqref{defcLtau} (see Remark \ref{rempencil}). Just like in the relaxed Allen-Cahn case (see Section 3 of \cite{LMPS16}), we shall prove that there is a one-to-one correspondence between the two sets, both in location and in multiplicities.
First observe that the family of operators ${\mathcal{L}}^\tau$ in \eqref{defcLtau} is defined for parameter values of $\tau > 0$ only, whereas the first order systems \eqref{Wsystem} are well defined for $\tau = 0$ as well. (This happens because the hyperbolic equation \eqref{hypAC} actually degenerates into a parabolic equation when $\tau \to 0^+$.) Thus, we shall prove the spectral equivalence between the two spectral problems assuming that $\tau > 0$. Notice that for each $\tau > 0$ the operator ${\mathcal{L}}^\tau : L^2 \times L^2 \to L^2 \times L^2$ is a closed, densely defined linear operator with domain ${\mathcal{D}}({\mathcal{L}}^\tau) = H^2 \times H^1$.
\begin{lemma}
\label{lemQinv}
For each $\lambda \in \mathbb{C}$ and $\tau > 0$, the mapping
\[
\begin{aligned}
{\mathcal{K}} : \ker ({\mathcal{L}}^\tau - \lambda) &\subset H^2 \times H^1 \, \longrightarrow \ker {\mathcal{T}}^\tau(\lambda) \subset H^1 \times H^1,\\
{\mathcal{K}} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} &:= \begin{pmatrix} v_1 \\ \partial_x v_1 \end{pmatrix}, \qquad \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \in \ker ({\mathcal{L}}^\tau - \lambda),
\end{aligned}
\]
is one-to-one and onto.
\end{lemma}
\begin{proof}
\smartqed
First we check that $(v_1,v_2)^\top \in \ker ({\mathcal{L}}^\tau - \lambda)$ implies that ${\mathcal{K}} (v_1,v_2)^\top \in \ker {\mathcal{T}}^\tau(\lambda)$. In that case we have the system
\[
\begin{aligned}
c \partial_x v_1 + v_2 &= \lambda v_1 \\
\tau^{-1} (\partial_x^2 + a(x)) v_1 + (c \partial_x - \tau^{-1} b(x)) v_2 &= \lambda v_2.
\end{aligned}
\]
Labeling $v := v_1$ and substituting the first equation into the second we immediately arrive at equation \eqref{specprob}, with $(v, v_x) \in H^1 \times H^1$. This shows that ${\mathcal{K}} (v_1, v_2)^\top = (v, v_x)^\top \in \ker {\mathcal{T}}^\tau(\lambda)$.
Now suppose that $(v, v_x)^\top \in \ker {\mathcal{T}}^\tau(\lambda) \subset H^1 \times H^1$. Then clearly $v \in H^2$ and let us define $v_1 := v$, $v_2 := \lambda v - cv_x$. It is then easy to verify that
\[
c \partial_x v_1 + v_2 = \lambda v = \lambda v_1, \qquad \text{and, }
\]
\[
\tau^{-1} (\partial_x^2 + a(x)) v_1 + (c\partial_x - \tau^{-1} b(x)) v_2 = \tau^{-1} (\lambda^2 \tau v - c \lambda \tau v_x) = \lambda (\lambda v - cv_x) = \lambda v_2.
\]
This yields $(v_1, v_2)^\top \in \ker ({\mathcal{L}}^\tau - \lambda)$. Thus, for each element $(v, v_x)^\top \in \ker {\mathcal{T}}^\tau(\lambda)$ there exists $(v_1, v_2)^\top \in \ker ({\mathcal{L}}^\tau - \lambda)$ such that $(v, v_x)^\top = {\mathcal{K}} (v_2, v_2)^\top$, and we verify that ${\mathcal{K}}$ is onto.
Finally, suppose that ${\mathcal{K}} (u_1, u_2)^\top = {\mathcal{K}}(v_1, v_2)^\top$ for $(u_1, u_2)$, $(v_1, v_2) \in \ker ({\mathcal{L}}^\tau - \lambda)$. This means that $(u_1, \partial_x u_1) = (v_1, \partial_x v_1)$ a.e. in $H^2 \times H^1$. But this implies that $v_2 = \lambda v_1 - c\partial_x v_1 = \lambda u_1 - c \partial_x u_1 = u_2$ a.e. in $H^1$ and we conclude that the mapping ${\mathcal{K}}$ is one-to one.
\qed
\end{proof}
An immediate consequence of the one-to-one correspondence between the kernels of ${\mathcal{L}}^\tau - \lambda$ and ${\mathcal{T}}^\tau(\lambda)$ is that the Fredholm properties of both operators are the same (see, e.g., Sandstede \cite{San02}, section 3.3). Therefore, if we naturally adopt Weyl's definition of spectra and define
\[
\begin{aligned}
\sigma_\mathrm{\tiny{pt}}({\mathcal{L}}^\tau) &:= \{ \lambda \in \mathbb{C}\,: \; {\mathcal{L}}^\tau - \lambda \,\text{ is Fredholm with index zero and has a} \\
& \qquad \qquad \qquad \text{non-trivial kernel} \},\\
\sigma_\mathrm{\tiny{ess}}({\mathcal{L}}^\tau) &:= \{ \lambda \in \mathbb{C}\,: \; {\mathcal{L}}^\tau - \lambda \,\text{ is either not Fredholm or has index
different } \\
& \qquad \qquad \qquad \text{from zero} \},
\end{aligned}
\]
with $\rho({\mathcal{L}}^\tau) = \mathbb{C} \backslash (\sigma_\mathrm{\tiny{pt}}({\mathcal{L}}^\tau) \cup \sigma_\mathrm{\tiny{ess}}({\mathcal{L}}^\tau))$, then we obtain the following
\begin{corollary}
\label{corsamespec}
For each $\tau > 0$,
\[
\sigma_\mathrm{\tiny{pt}} = \sigma_\mathrm{\tiny{pt}}({\mathcal{L}}^\tau), \quad \sigma_\mathrm{\tiny{ess}} = \sigma_\mathrm{\tiny{ess}}({\mathcal{L}}^\tau), \quad \rho = \rho({\mathcal{L}}^\tau),
\]
where the sets on the left hand sides of the above equalities are, of course, the sets of Definition \ref{defsigmatwo}.
\end{corollary}
For $\lambda$ in the point spectrum, it is clear from Lemma \ref{lemQinv} that the dimensions of the finite-dimensional kernels are the same and, hence, the geometric multiplicity of $\lambda$ remains the same. Moreover, the mapping ${\mathcal{K}}$ can also be used to show that the Jordan block structures of ${\mathcal{L}}^\tau - \lambda$ and ${\mathcal{T}}^\tau(\lambda)$ coincide, that is, the algebraic multiplicity (the length of each maximal Jordan chain) is the same whether computed for one operator or for the other.
\begin{proposition}
\label{propJordan}
The mapping ${\mathcal{K}}$ induces a one-to-one correspondence between Jordan chains.
\end{proposition}
\begin{proof}
\smartqed
Suppose $(\varphi, \psi)^\top \in \ker ({\mathcal{L}}^\tau - \lambda)$. This implies the following system of equations,
\[
\begin{aligned}
c \varphi_x + \psi &= \lambda \varphi,\\
\tau^{-1} (\partial_x^2 + a(x)) \varphi + (c \partial_x - \tau^{-1} b(x)) \psi &= \lambda \psi.
\end{aligned}
\]
Take the next element in a Jordan chain, say, $(v_1, v_2)^\top \in H^2 \times H^1$ such that
\[
({\mathcal{L}}^\tau - \lambda) \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} \varphi \\ \psi \end{pmatrix}.
\]
This yields
\[
\begin{aligned}
c \partial_x v_1 + v_2 - \lambda v_1 &= \varphi,\\
\tau^{-1} (\partial_x^2 + a(x)) v_1 + (c\partial_x - \tau^{-1} b(x)) v_2 - \lambda v_2 &= \psi.
\end{aligned}
\]
Notice that ${\mathcal{K}} (v_1, v_2)^\top = (v_1, \partial_x v_1)^\top$, ${\mathcal{K}}(\varphi, \psi)^\top = (\varphi, \varphi_x)^\top$. Now substitute $\psi = \lambda \varphi - c \varphi_x$ and $v_2 = \varphi + \lambda v_1 - c\partial_x v_1$ in order to obtain a scalar equation for $v_1$ and $\varphi$. The result is
\[
\tau^{-1} (\partial_x^2 + a(x)) v_1 + (c\partial_x - \tau^{-1} b(x) - \lambda) (\varphi + \lambda v_1 - c\partial_x v_1) = \lambda \varphi - c \varphi_x.
\]
Labeling $v := v_1$, last equation reads
\[
(1-c^2 \tau) v_{xx} + (cb(x) + 2 \tau \lambda) v_x - (\lambda^2 \tau v + \lambda b(x) - a(x))v = (b(x) + 2 \tau \lambda) \varphi - 2c\tau \varphi_x,
\]
which is equivalent to
\[
(\partial_x - \mathbb{A}^\tau(x,\lambda)) \begin{pmatrix} v \\ v_x \end{pmatrix} = \Big( \mathbb{A}_1^\tau(x) + 2 \lambda \mathbb{A}^\tau_2(x) \Big) \begin{pmatrix} \varphi \\ \varphi_x \end{pmatrix}.
\]
Generalizing this procedure, we observe that solutions to
\[
({\mathcal{L}}^\tau - \lambda) \begin{pmatrix} v_1^j \\ v_2^j \end{pmatrix} = \begin{pmatrix} v_1^{j-1} \\ v_2^{j-1} \end{pmatrix},
\]
for some $j \geq 1$, are in one-to-one correspondence to solutions to
\[
{\mathcal{T}}^\tau(\lambda) {\mathcal{K}} \begin{pmatrix} v_1^j \\ v_2^j \end{pmatrix} = (\partial_\lambda \mathbb{A}^\tau(x,\lambda)) {\mathcal{K}} \begin{pmatrix} v_1^{j-1} \\ v_2^{j-1} \end{pmatrix}.
\]
We conclude that a Jordan chain for the operator ${\mathcal{L}}^\tau - \lambda$ induces a Jordan chain for ${\mathcal{T}}^\tau(\lambda)$ with the same block structure and length.
\qed
\end{proof}
\begin{corollary}
\label{corsameptsp}
Assume $\tau > 0$. Then for any complex number $ \lambda \in \mathbb{C}$ there holds
\[
\lambda \in \sigma_\mathrm{\tiny{pt}} \quad \text{if and only if} \quad \lambda \in \sigma_\mathrm{\tiny{pt}}({\mathcal{L}}^\tau),
\]
with the same algebraic and geometric multiplicities (here $\sigma_\mathrm{\tiny{pt}}$ is the set in Definition \ref{defsigmatwo}).
\end{corollary}
\begin{remark}
The results of Corollary \ref{corsamespec} and Proposition \ref{propJordan} generalize the spectral equivalence proved in the relaxed Allen-Cahn case (see Section 3 in \cite{LMPS16}). It is remarkable, however, that for the Allen-Cahn model with relaxation the associated matrix ${\mathcal{L}}^\tau$ is a first order differential operator, whereas in the present (general) case the operator is of second order.
\end{remark}
\section{Asymptotic limits and the essential spectrum}
\label{secess}
In this section we analyze the asymptotic equations
\begin{equation}
\label{Wasymp}
W_x = \mathbb{A}^\tau_\pm(\lambda) W,
\end{equation}
wherupon the asymptotic coefficients are defined in \eqref{asympcoeff}, and
which will allow us, in turn, to
locate the essential spectrum of our problem.
\subsection{The asymptotic equations}
Take a look at the asymptotic coefficients \eqref{asympcoeff}. Let us denote the characteristic polynomial of
$\mathbb{A}^\tau_\pm(\lambda)$ as
\begin{equation}
\label{defcharpol}
p_\pm^\tau(\mu) = \det \big( \mathbb{A}_\pm^\tau (\lambda) - \mu I\big).
\end{equation}
Notice that $\mu$ is a root of $p_\pm^\tau(\mu) = 0$ if and only if $\kappa = (1-c^2 \tau) \mu$ is a root of
\[
\begin{aligned}
\det \big(\kappa I - (1-c^2 \tau) \mathbb{A}_\pm^\tau (\lambda)\big) &= \det \begin{pmatrix}
\kappa & & -(1-c^2\tau) \\ - \tau
\lambda^2 - \lambda b_\pm - |a_\pm| & & \, \kappa + c(b_\pm + 2 \tau \lambda)
\end{pmatrix}\\
&= \kappa^2 + \kappa c(b_\pm + 2 \tau \lambda) - (1 - c^2 \tau)(\tau \lambda^2 + \lambda b_\pm + |a_\pm|) \\
&= 0.
\end{aligned}
\]
Suppose that $\kappa = i \xi$, with $\xi \in \mathbb{R}$. Then the $\lambda$-roots of the equation
\begin{equation}
\label{disprel}
\xi^2 - ic\xi (b_\pm + 2 \tau \lambda) + (1-c^2\tau)(\tau \lambda^2 + b_\pm \lambda + |a_\pm|) = 0,
\end{equation}
define algebraic curves in the complex plane, bounding the essential spectrum. We denote these curves as
\begin{equation}
\label{algcurves}
\lambda = \lambda_{1,2}^\pm(\xi), \qquad \xi \in \mathbb{R}.
\end{equation}
Equation \eqref{disprel} is the \textit{dispersion relation} for the wave solutions to the constant
coefficient asymptotic equations.
\begin{remark}
It is clear that $\lambda = 0$ does not belong to any of the algebraic curves \eqref{algcurves}, inasmuch as
$\xi^2 - ic\xi b_\pm + (1-c^2\tau) |a_\pm|$ has strictly positive real part for all $\xi \in \mathbb{R}$.
\end{remark}
\subsubsection{The case $\tau = 0$}
We first analyze these curves in the case when $\tau = 0$. Then the dispersion relation \eqref{disprel} reads
\[
\xi^2 - ic\xi b_\pm + b_\pm \lambda + |a_\pm| = 0,
\]
and the single root is simply
\begin{equation}
\label{algcurvtau0}
\lambda_0^\pm(\xi) = -b_\pm^{-1}|a_\pm| + ic\xi - b_\pm \xi^2,
\end{equation}
for all $\xi \in \mathbb{R}$. These curves lie on the stable half plane with $\Re \lambda < 0$. In fact, there exist
\begin{equation}
\chi_0^\pm = \tfrac{1}{2}b_\pm^{-1}|a_\pm|,
\end{equation}
\[
\chi_0 = \min \{\chi_0^+, \chi_0^-\} > 0
\]
such that
\[
\Re \lambda_0^\pm(\xi) < - \chi_0^\pm \leq -\chi_0 < 0, \qquad \text{for all} \; \xi \in \mathbb{R}.
\]
In other words, there is a \textit{spectral gap}.
\subsubsection{The case $\tau > 0$}
We now examine the case when $\tau > 0$. Recall that $0 < \tau < 1/c^2$ thanks to the subcharacteristic
condition. Let us suppose that $\lambda(\xi)$ belongs to one of the curves \eqref{algcurves} and let
$\eta(\xi) = \Re \lambda(\xi)$, $\beta(\xi) = \Im \lambda(\xi)$. Then, take the real and imaginary parts of
the dispersion relation \eqref{disprel} to obtain
\begin{equation}
\label{realp}
\xi^2 + 2c\tau \xi \beta + (1-c^2\tau)\big( \tau(\eta^2 - \beta^2) + \eta b_\pm + |a_\pm|\big) = 0.
\end{equation}
\begin{equation}
\label{imagp}
-c\xi b_\pm - 2c\tau \xi \eta + (1-c^2\tau)\big( 2\tau \eta\beta + b_\pm \beta\big) = 0.
\end{equation}
\begin{remark}
Upon inspection of \eqref{realp} and \eqref{imagp} we notice that if we assume that $\eta = \Re \lambda = 0$
for some $\xi \in \mathbb{R}$ then $-c\xi b_\pm + (1-c^2\tau) \beta b_\pm = 0$. Since $b_\pm > 0$ this implies that
$\beta = c\xi/(1-c^2\tau)$. Substituting into \eqref{realp} we obtain $\xi^2 + \tau c^2 \xi^2 /(1-c^2\tau) +
|a_\pm| = 0$, which is a contradiction with $|a_\pm| > 0$, $\tau > 0$, $1-c^2\tau > 0$. This shows that the
algebraic curves never cross the imaginary axis; they remain in either the stable or the unstable complex half
plane.
\end{remark}
Notice that equation \eqref{imagp} can be written as
\[
\big( \beta - \frac{c\xi}{1-c^2\tau}\big)(b_\pm + 2\tau \eta) = 0.
\]
Thus, either
\begin{align}
\eta(\xi) &= - \frac{b_\pm}{2\tau}, \label{casei}\\
\textrm{or, } \;\; \beta(\xi) &= \frac{c\xi}{1-c^2\tau}. \label{caseii}
\end{align}
First, let us consider case \eqref{casei}. Substituting into \eqref{realp} yields
\begin{equation}
\label{eqforbeta}
\tau \beta^2 - \big( \frac{2c\tau \xi}{1-c^2\tau}\big) \beta - |a_\pm| + \frac{b_\pm^2}{4\tau} -
\frac{\xi^2}{1-c^2\tau} = 0.
\end{equation}
This equation has real solutions $\beta$ provided that
\[
\Delta_1 := \frac{4c^2 \tau^2 \xi^2}{(1-c^2 \tau)^2} - 4\tau \Big( -|a_\pm| + \frac{b_\pm^2}{4\tau} -
\frac{\xi^2}{1-c^2\tau}\Big) \geq 0,
\]
or equivalently,
\begin{equation}
\label{star}
\xi^2 (1-c^2 \tau)^{-2} + |a_\pm| \geq \frac{b_\pm^2}{4 \tau}.
\end{equation}
On the other hand, if we consider case \eqref{caseii} then after substituting into \eqref{realp} we obtain
\begin{equation}
\label{eqforeta}
\tau \eta^2 + b_\pm \eta + |a_\pm| + \frac{\xi^2}{(1-c^2\tau)^2} = 0.
\end{equation}
Last equation has real solutions $\eta$ if and only if
\[
\Delta_2 := b_\pm^2 - 4\tau \Big( |a_\pm| + \frac{\xi^2}{(1-c^2\tau)^2} \Big) \geq 0,
\]
that is, when
\begin{equation}
\label{dstar}
\xi^2(1-c^2\tau)^{-2} + |a_\pm| \leq \frac{b_\pm^2}{4\tau}.
\end{equation}
Therefore, clearly, $\mathrm{sgn}\, \Delta_2 = - \mathrm{sgn}\, \Delta_1$. We consider two cases:\\
\noindent \textit{Case (I):} Suppose that for a certain parameter value $\tau > 0$ there holds
\begin{equation}
\label{taularge}
\frac{b_\pm^2}{4\tau} < |a_\pm|,
\end{equation}
which means that for both the asymptotic states, or for one of them, $\tau > 0$ is sufficiently large such
that \eqref{taularge} is true.
\begin{remark}
It is to be observed that this case happens in the example when $g \equiv 1$, $f(u) = u(1-u)(u-1/2)$ if we
take $\tau = 1$, yielding $b_\pm = 1$, $|a_\pm| = 1/2$.
\end{remark}
Whence, if \eqref{taularge} holds then condition \eqref{dstar} is never satisfied and \eqref{star} is always
true. Therefore there are only real solutions for $\beta$ in \eqref{eqforbeta} inasmuch as $\Delta_1 > 0$ for
all $\xi \in \mathbb{R}$. This implies that the only algebraic curve solutions $\lambda = \lambda(\xi)$ to
\eqref{disprel} are
\begin{equation}
\label{etabeta}
\begin{aligned}
\Re \lambda (\xi) = \eta (\xi) &= - \frac{b_\pm}{2\tau}, \\
\Im \lambda (\xi) = \beta(\xi) &= \frac{c\xi}{1-c^2\tau} \pm \frac{1}{2\tau} \sqrt{\Delta_1(\xi)},
\end{aligned}
\end{equation}
for all $\xi \in \mathbb{R}$. Notice that there exists $\chi_1^\pm(\tau) := b_\pm/(4\tau) > 0$ such that there is a
spectral gap:
\[
\Re \lambda(\xi) < - \chi_1^\pm < 0, \qquad \xi \in \mathbb{R}.
\]
\noindent \textit{Case (II):} Now suppose that for certain parameter values
\begin{equation}
\label{tausmall}
\frac{b_\pm^2}{4\tau} \geq |a_\pm|.
\end{equation}
\begin{remark}
Notably, this case occurs for systems of Cattaneo-Maxwell type with $f(u) = u(1-u)(u-\alpha)$, $g(u,\tau) = 1
- \tau f'(u)$, $\alpha \in (0,1)$. Here $g(u,\tau) > 0$ provided that
\[
0 < \tau < \frac{3}{1-\alpha + \alpha^2},
\]
as the reader may easily verify. (This warrants hypothesis \eqref{H1} to hold.) Since $g(0,\tau) = b_- = 1 +
\tau \alpha > 0$, $g(1,\tau) = b_+ = 1+\tau(1-\alpha) > 0$, then clearly
\begin{align*}
\frac{b_-^2}{4\tau} &= \frac{(1+\alpha\tau)^2}{4\tau} \geq \alpha = |a_-|,\\
\frac{b_+^2}{4\tau} &= \frac{(1+(1-\alpha)\tau)^2}{4\tau} \geq 1-\alpha = |a_+|,
\end{align*}
verifying the two conditions \eqref{tausmall}.
\end{remark}
Assuming \eqref{tausmall}, let $\xi_0^\pm \geq 0$ be the nonnegative solution to
\[
(\xi_0^\pm)^2 = (1-c^2\tau)^2\big( \frac{b_\pm^2}{4\tau} - |a_\pm|\big).
\]
Henceforth, for every $\xi \in (-\xi_0^\pm,\xi_0^\pm)$ we have that
\[
\xi^2 < (1-c^2\tau)^2\big( \frac{b_\pm^2}{4\tau} - |a_\pm|\big),
\]
condition \eqref{dstar} is satisfied, and consequently, $\Delta_2(\xi) > 0$. In that range for $\xi$ the
solutions for $\beta$ and $\eta$ are thus given by
\[
\beta(\xi) = \frac{c\xi}{1-c^2\tau}, \qquad \xi \in (-\xi_0^\pm,\xi_0^\pm),
\]
and by
\begin{equation}
\label{etagood}
\eta(\xi) = \frac{1}{2\tau} \big( b_\pm \pm \sqrt{\Delta_2(\xi)}\big), \qquad \xi \in (-\xi_0^\pm,\xi_0^\pm),
\end{equation}
respectively. Observe, however, that $\Delta_1(\xi), \Delta_2(\xi) \to 0$ as $|\xi| \uparrow \xi_0^\pm$; that
$\beta(\xi) \to \pm c\xi_0/(1-c^2\tau)$ as $\xi \to \pm \xi_0^\pm$, $|\xi| < \xi_0^\pm$; and that $\eta(\xi)
\to -b_\pm/2\tau$ as $|\xi| \uparrow \xi_0^\pm$. This behavior guarantees the continuity of the algebraic
curves at $|\xi| = \xi_0^\pm$, because the roots of equation \eqref{eqforbeta} at $|\xi|=\xi_0^\pm$ are
\[
\beta(\xi_0) = \frac{\pm c\xi_0}{1-c^2\tau}
\]
(as $\Delta_1(\xi_0^\pm) = 0$), and $\eta$ is constant, given by $\eta = -b_\pm/2\tau$. Therefore, for values
$|\xi| \geq \xi_0^\pm$, $\Delta_1$ and $\Delta_2$ switch signs, $\Delta_1$ is now positive and the solutions
for $\eta$ and $\beta$ are given by formulas \eqref{etabeta}.
Closer inspection of \eqref{etagood} reveals that
\[
\eta(\xi) = - \frac{b_\pm}{2\tau} \pm \sqrt{\frac{b_\pm^2}{4\tau^2} - \frac{1}{\tau}\Big( |a_\pm| +
\frac{\xi^2}{(1-c^2\tau)^2}\Big)} \; \leq \, - \frac{b_\pm}{2\tau} + \sqrt{\frac{b_\pm^2}{4\tau^2} -
\frac{|a_\pm|}{\tau}} \; < 0,
\]
for all $|\xi| \leq \xi_0^\pm$. Therefore, in case (II) there exists
\[
\chi_2^\pm(\tau) = \frac{b_\pm}{4\tau} - \frac{1}{2} \sqrt{\frac{b_\pm^2}{4\tau^2} - \frac{|a_\pm|}{\tau}}
>0,
\]
such that
\[
\Re \lambda(\xi) < - \chi_2^\pm < 0, \qquad |\xi| \leq \xi_0^\pm,
\]
and there is also a spectral gap.
Under these considerations we now define, for each fixed $\tau \geq 0$,
\begin{equation}
\label{defspectralgap}
0 < \chi_0^\pm (\tau) := \begin{cases}
\tfrac{1}{2}b_\pm^{-1}|a_\pm|, & \text{if } \; \tau = 0,\\
\displaystyle{\tfrac{1}{2}\Big( \frac{b_\pm}{2\tau} - \sqrt{\frac{b_\pm^2}{4\tau^2} -
\frac{|a_\pm|}{\tau}} \, \Big)}, & \text{if } \; b_\pm^2 \geq 4\tau |a_\pm|, \, \tau > 0,\\
\displaystyle{\frac{b_\pm}{4\tau}}, & \text{otherwise.}
\end{cases}
\end{equation}
Thus we have proved the following
\begin{lemma}[spectral gap]
\label{lemspectralgap}
For each $\tau \geq 0$, there exists a uniform
\begin{equation}
\label{defchi0}
\chi_0(\tau) = \min \{\chi_0^+(\tau), \chi_0^-(\tau)\} > 0,
\end{equation}
(where $\chi_0^\pm (\tau)$ are defined in \eqref{defspectralgap}) such that the algebraic curves $\lambda =
\lambda_{1,2}^\pm(\xi)$, $\xi \in \mathbb{R}$, solutions to the dispersion relations \eqref{disprel}, satisfy
\begin{equation}
\label{spectralgapeq}
\mathrm{Re}\, \lambda_{1,2}^\pm(\xi) < - \chi_0(\tau) < 0, \qquad \xi \in \mathbb{R}.
\end{equation}
\end{lemma}
\begin{remark}
The significance of Lemma \ref{lemspectralgap} is that there is no accumulation of essential spectrum at the
eigenvalue $\lambda = 0$, which is an isolated eigenvalue with finite multiplicity (see Lemma \ref{lemalgm}
below).
Notice that for each
finite $\tau \geq 0$, the bound $\chi_0(\tau)$ is positive. There could be accumulation of the essential
spectrum in the case when $\tau \to +\infty$ (for which, it may happen, that $\chi_0(\tau) \to 0$), but that
case is precluded by our hypothesis \eqref{H2}, with an upper bound $\tau < \tau_m < +\infty$. In the case
of the relaxation model with Cattaneo-Maxwell transfer law (see equation \eqref{ACrelax}), the parameter
values are bounded by a characteristic relaxation time associated to the reaction, $\tau_m = 1/\max_{u \in
[0,1]} |f'(u)|$.
\end{remark}
\subsection{Hyperbolicity and consistent splitting}
For a given $\tau \geq 0$, we define the following open, connected region of the complex plane,
\begin{equation}
\label{defOmega}
\Omega := \{\lambda \in \mathbb{C} \, : \, \Re \lambda > - \chi_0(\tau)\}.
\end{equation}
It properly contains the unstable complex half plane $\mathbb{C}_+ = \{ \Re \lambda > 0\}$. This is called the
region of consistent splitting \cite{San02}. Denote
$S^\tau_\pm(\lambda)$ and $U^\tau_\pm(\lambda)$ as the stable and unstable eigenspaces of
$\mathbb{A}^\tau_\pm(\lambda)$, respectively.
\begin{lemma}\label{lemconsplit}
Given $\tau \geq 0$, for all $\lambda \in \Omega$ the coefficient matrices $\mathbb{A}^\tau_\pm(\lambda)$ have no
center eigenspace and, moreover,
\[
\dim S^\tau_\pm(\lambda) = \dim U^\tau_\pm(\lambda) = 1.
\]
\end{lemma}
\begin{proof}
\smartqed
Take $\lambda \in \Omega$ and suppose $\kappa = i \xi$, with $\xi \in \mathbb{R}$, is an eigenvalue of
$\mathbb{A}_\pm^\tau(\lambda)$. Then $\lambda$ belongs to one of the algebraic curves \eqref{algcurves}. But
\eqref{spectralgapeq} yields a contradiction with $\lambda \in \Omega$. Therefore, the matrices
$\mathbb{A}_\pm^\tau(\lambda)$ have no center eigenspace.
Since $\Omega$ is a connected region of the complex plane, it suffices to compute the dimensions of
$S_\pm^\tau(\lambda)$ and $U_\pm^\tau(\lambda)$ when $\lambda = \eta \in \mathbb{R}_+$, sufficiently large. $\mu$ is
a root of $p_\pm^\tau(\mu) = \det (\mathbb{A}_\pm^\tau(\lambda) - \mu) = 0$ if and only if $\kappa = (1-c^2 \tau)
\mu$ is a solution to
\begin{equation}
\label{eqkappa}
\kappa^2 + \kappa c(b_\pm + 2 \tau \lambda) - (1 - c^2 \tau)(\tau \lambda^2 + \lambda b_\pm + |a_\pm|) = 0.
\end{equation}
Assuming $\lambda = \eta \in \mathbb{R}_+$, the roots are
\[
\kappa = - \frac{c}{2}(b_\pm + 2 \tau \eta) \pm \frac{1}{2} \sqrt{c^2 (b_\pm + 2\tau \eta)^2 +
4(1-c^2\tau)(\tau \eta^2 + \eta b_\pm + |a_\pm|)}.
\]
Clearly, for each $\eta > 0$, one of the roots is positive and the other is negative. This proves the lemma.
\qed \end{proof}
The most important consequence of last lemma is the following
\begin{corollary}[stability of the essential spectrum]\label{corstabess}
For each $\tau \geq 0$, the essential spectrum is contained in the stable half-plane. More precisely,
\[
\sigma_\mathrm{\tiny{ess}} \subset \{\lambda \in \mathbb{C} \, : \, \mathrm{Re} \, \lambda \leq - \chi_0(\tau) < 0\}.
\]
\end{corollary}
\begin{proof}
\smartqed
The proof follows standard arguments \cite{KaPro13}. Fix $\lambda \in \Omega$. Since $\mathbb{A}_\pm^\tau(\lambda)$
are hyperbolic, by exponential dichotomies
theory (cf. Coppel \cite{Cop78}, Sandstede
\cite{San02}) the asymptotic systems $W_x = \mathbb{A}_\pm^\tau(\lambda)W$ have exponential dichotomies in $x \in
\mathbb{R}_+ = (0,+\infty)$ and in $x \in \mathbb{R}_- = (-\infty,0)$, respectively,
with Morse indices
\[
\begin{aligned}
i_+(\lambda) &= \dim U_+^\tau(\lambda) = 1, \\
i_-(\lambda) &= \dim U_-^\tau(\lambda) = 1.
\end{aligned}
\]
This implies (cf. Palmer \cite{Pal1,Pal2}, Sandstede \cite{San02}), that the variable coefficient operators
${\mathcal{T}}^\tau(\lambda)$ are Fredholm as well, with index
\[
\text{ind}\, {\mathcal{T}}^\tau(\lambda) = i_+(\lambda) - i_-(\lambda) =
0,
\]
showing that $\Omega \subset \mathbb{C}\backslash \sigma_\mathrm{\tiny{ess}}$, or equivalently, that $\sigma_\mathrm{\tiny{ess}} \subset \mathbb{C}\backslash
\Omega = \{\Re
\lambda \leq - \chi_0(\tau)\}$, as claimed.
\qed \end{proof}
\begin{corollary}\label{corsplit}
For every $\lambda \in \Omega$, the eigenvalues of the asymptotic coefficients \eqref{asympcoeff} are given
by
\begin{equation}
\label{evaluesmu}
\mu^\pm_{1,2}(\lambda) = - \frac{c}{2(1-c^2 \tau)}(b_\pm + 2 \tau \lambda) + \omega^\pm_{1,2}(\lambda),
\end{equation}
whereupon
\[
\omega_1^\pm (\lambda) := - \frac{1}{2} \Theta_{\pm}(\lambda)^{1/2}, \qquad \omega_2^\pm (\lambda) :=
\frac{1}{2} \Theta_{\pm}(\lambda)^{1/2},
\]
and,
\[
\Theta_{\pm}(\lambda) = (1-c^2\tau)^{-2} \Big( c^2 b_\pm^2 + 4(\tau \lambda^2 + b_\pm \lambda + (1-c^2
\tau)|a_\pm|) \Big).
\]
Morever, for every $\lambda \in \Omega$,
\[
\mathrm{Re} \, \mu_1^\pm(\lambda) < 0 < \mathrm{Re} \, \mu_2^\pm(\lambda),
\]
that is, $\mu_1^+(\lambda)$ is the decaying mode at $+\infty$, and $\mu_2^-(\lambda)$ is the decaying mode at
$-\infty$.
\end{corollary}
\begin{proof}
\smartqed
Since $p_\pm^\tau(\mu) = 0$ if and only if $\kappa = (1-c^2 \tau)\mu$ is a root of the characteristic
equation \eqref{eqkappa}, then it is clear that for each $\lambda \in\Omega$ the eigenvalues of
$\mathbb{A}_\pm^\tau(\lambda)$ are given by \eqref{evaluesmu}. A little algebra yields the expression for the
discriminant $\Theta_\pm(\lambda)$, an analytic function of $\lambda$. From the proof of Lemma
\ref{lemconsplit}, we know that, for $\lambda \in \mathbb{R}$ and $\lambda \gg 1$, the only eigenvalue with
negative real part is $\mu_1^\pm(\lambda)$. Since $\Omega$ is connected and the eigenvalues are continuous
(analytic) in $\lambda$, we conclude that $\Re \mu_1^\pm(\lambda) < 0$ for all $\lambda \in \Omega$
(otherwise, the hyperbolicity, and consequently the consistent splitting, would be violated). The same argument
applies to $\mu_2^\pm(\lambda)$ and the conclusion follows.
\qed
\end{proof}
\section{Point spectral stability}
\label{secptsp}
This section is devoted to showing that the point spectrum is stable. The proof presented here makes use of
energy estimates and contrasts with the one reported in \cite{LMPS16} for the particular case of the
Allen-Cahn model with relaxation. The former proof was based on a perturbation argument in the vicinity of
$\tau = 0$ and a further extension to the whole parameter domain. In contrast, here we perform
energy estimates in the frequency regime that require to apply a transformation
on the $H^2$-eigenfunction. Thanks to its decaying behaviour, the transformed eigenfunction also belongs to
$H^2$ and we are able to perform the energy estimates on the new spectral equation. We close the section by
showing that the eigenvalue $\lambda = 0$ is simple and by stating the main result of the paper.
\subsection{Decay of solutions to spectral equations}
\begin{lemma}
\label{goodw}
Suppose $v \in H^2$ is a solution to the spectral equation \eqref{specprob} for some $\lambda \in
\sigma_\mathrm{\tiny{pt}}$ with $\mathrm{Re}\, \lambda \geq 0$ and $\lambda \in \Omega$. If we define
\begin{equation}
\label{transfw}
w(x) = \exp \left( \frac{c}{2(1-c^2 \tau)} \int_{x_0}^x b(s) \, ds \right) v(x), \qquad x \in \mathbb{R},
\end{equation}
then $w \in H^2$. Here $x_0 \in \mathbb{R}$ is fixed but arbitrary.
\end{lemma}
\begin{proof}
\smartqed
Since $\lambda \in \sigma_\mathrm{\tiny{pt}}$ there exists $W = (v,v_x)^\top \in H^1 \times H^1$ such that ${\mathcal{T}}^\tau(\lambda) W
= 0$. This implies, in turn, that $v \in H^2$ is a solution to the spectral equation \eqref{specprob}. To
analyze the decaying properties of $v$ (equivalently, of $W$) we invoke the Gap Lemma \cite{GZ98,KS98}, which
relates the decaying properties of the solutions to the variable coefficient system \eqref{Wsystem} to those
of the solutions of the constant coefficient systems \eqref{Wasymp}, provided that $\mathbb{A}^\tau(x,\lambda)$
approaches $\mathbb{A}_\pm^\tau(\lambda)$ exponentially fast as $x \to \pm \infty$. For the precise statement of the
Gap Lemma we refer the reader to Lemma A.11 in \cite{Zum04}, or Appendix C in \cite{MZ02}.
Suppose that $c > 0$. Since $b > 0$, it is clear that if $x < x_0$ then $|w| \leq |v|$ and $w$ decays like
$v$ as $x \to - \infty$ Thus, we need to make precise the decaying behaviour of $v$ as $x \to +\infty$. By
exponential decay of the profile \eqref{expdec}, it is clear that
\[
|\mathbb{A}^\tau(x,\lambda) - \mathbb{A}_\pm^\tau(\lambda)| \leq C e^{-\nu |x|},
\]
as $x \to \pm \infty$, for some $C, \nu > 0$, uniformly in $\lambda$. Then, applying the Gap Lemma and
Corollary \ref{corsplit}, the decaying solution $W$ at $+\infty$ to the variable coefficient equation behaves
as
\[
W(x,\lambda) = e^{\mu_1^+(\lambda)} \Big( V_1^+(\lambda) + O(e^{-\nu|x|} |V_1^+(\lambda)|) \Big), \quad x >
0,
\]
where $V_1^+(\lambda)$ is the eigenvector of $\mathbb{A}_\pm^\tau(\lambda)$ associated to the eigenmode
$\mu_1^+(\lambda)$. This imples that $v$ and $v_x$ decay, at most, as
\[
|v|, |v_x| \leq C e^{\Re \mu_1^+(\lambda) x},
\]
as $x \to +\infty$. We then readily see, from Corollary \ref{corsplit}, that
\[
\begin{aligned}
|w| &\leq C \exp \Big( \frac{c}{2(1-c^2 \tau)} \int_{x_0}^x |b(s) - b_+| \, ds \Big) \times \\ & \qquad
\times \exp \Big( \Big( - \frac{c \tau \Re \lambda}{2(1-c^2 \tau)} - \frac{1}{2 \sqrt{2}} \sqrt{\Re
\Theta_+(\lambda) + |\Theta_+(\lambda)|}\Big) x \Big) \\
&\leq C \exp \Big( - \frac{c \tau (\Re \lambda)x}{2(1-c^2 \tau)}\Big) \exp \Big( - \frac{x}{2 \sqrt{2}}
\sqrt{\Re \Theta_+(\lambda) +
|\Theta_+(\lambda)|} \Big) \, \to 0,
\end{aligned}
\]
as $x \to +\infty$, thanks to exponential decay of the profile, which yields
\[
\exp \Big( \frac{c}{2(1-c^2 \tau)} \int_{x_0}^x |b(s) - b_\pm| \, ds \Big) \leq C \exp( -e^{-\nu x}) \leq C.
\]
This shows that $w$ decays exponentially fast as $x \to +\infty$. Since $v_x$ decays as the same rate as $v$,
it is easy to verify that $w_x$ also decays exponentially fast at $+\infty$. We conclude that $w \in H^1$.
Upon differentiation one can prove that, in fact, $w \in H^2$, as $w_{xx}$ decays exponentially fast as well
at $+\infty$. Details are left to the dedicated reader.
The case $c < 0$ can be treated similarly, inasmuch as the decay at $-\infty$ of the eigenfunction $W =
(v,v_x)^\top$ is determined by the eigenmode $\mu_2^-(\lambda)$; an analogous argument applies. This
concludes the proof of the lemma.
\qed \end{proof}
\subsection{Energy estimates}
Suppose that $\lambda \in \sigma_\mathrm{\tiny{pt}}$, with $\Re \lambda \geq 0$ (and consequently, $\lambda \in \Omega$). Then
there exists $W = (v, v_x)^\top \in H^1 \times H^1$ such that
${\mathcal{T}}^\tau (\lambda) W = 0$. This is tantamount to have an $H^2$ solution $v$ to the spectral equation
\eqref{specprob}. Consider the transformation
\[
v(x) = w(x) e^{\theta(x)},
\]
where the function $\theta = \theta(x)$ is to be determined. Upon substitution into
\eqref{specprob} we obtain
\[
\begin{aligned}
\lambda^2 \tau w - 2c\lambda \tau (w_x + \theta_x w) + \lambda b(x) w &= (1 - c^2 \tau) w_{xx} + \big(
2(1-c^2 \tau) \theta_x + c b(x) \big) w_x + \\
& \; + \big( (1-c^2 \tau) (\theta_x^2 + \theta_{xx}) + cb(x) \theta_x + a(x) \big) w.
\end{aligned}
\]
Choose $\theta$ such that
\[
\theta_x = - \frac{c}{2(1-c^2\tau)} b(x).
\]
This yields
\begin{equation}
\label{eqsix}
\lambda^2 \tau w - 2c \lambda \tau w_x + \frac{\lambda b(x)}{1-c^2 \tau} w = (1 - c^2 \tau) w_{xx} + H(x) w,
\end{equation}
whereupon
\[
H(x) := a(x) - \frac{c^2 b(x)^2}{4(1-c^2 \tau)} - \tfrac{1}{2} c b'(x).
\]
If we apply the same procedure to the eigenfunction $U_x \in H^2$ associated to the eigenvalue $\lambda = 0
\in \sigma_\mathrm{\tiny{pt}}$, denoting $U_x = \psi e^\theta$ we arrive at
\begin{equation}
\label{eqsixpsi}
0 = (1 - c^2 \tau) \psi_{xx} + H(x) \psi.
\end{equation}
By monotonicity of the profile, $U_x > 0$, we know that $\psi > 0$ and we can solve for $H$ in
\eqref{eqsixpsi}, yielding
\[
H(x) = - (1-c^2 \tau) \frac{\psi_{xx}}{\psi}.
\]
Substituting back into \eqref{eqsix} we obtain
\begin{equation}
\label{eqsix2}
\lambda^2 \tau w - 2c \lambda \tau w_x + \frac{\lambda b(x)}{1-c^2 \tau} w = (1 - c^2 \tau) \Big( w_{xx} -
\frac{\psi_{xx}}{\psi} w \Big).
\end{equation}
Notice that thanks to Lemma \ref{goodw}, we have that this is an spectral equation for $w \in H^2$. We
perform standard energy estimates on equation \eqref{eqsix2}. Multiply by $\overline{w}$ and integrate by
parts in $\mathbb{R}$. The result is,
\[
\begin{aligned}
\lambda \tau^2 \| w \|_{L^2}^2 - 2c \lambda \tau \int_{\mathbb{R}} \overline{w} w_x \, dx &+ \frac{\lambda}{1-c^2
\tau} \int_{\mathbb{R}} b(x) |w|^2 \, dx = \\ &= (1-c^2 \tau) \left( - \int_{\mathbb{R}} |w_x|^2 \, dx + \int_{\mathbb{R}} \psi_x
\partial_x \Big( \frac{|w|^2}{\psi} \Big) \, dx \right)
\end{aligned}
\]
Using the identity
\begin{equation*}
\psi^{2}\left|\left(\frac{w}{\psi} \right)_x \right|^{2} = - \left( \psi_x \left(\frac{|w|^{2}}{\psi}
\right)_x-|w_x|^{2} \right),
\end{equation*}
and substituting, we obtain the estimate
\begin{equation}
\label{basicee}
\begin{aligned}
\lambda \tau^2 \| w \|_{L^2}^2 - 2c \lambda \tau \int_{\mathbb{R}} \overline{w} w_x \, dx &+ \frac{\lambda}{1-c^2
\tau} \int_{\mathbb{R}} b(x) |w|^2 \, dx = \\ &= - (1-c^2 \tau) \int_{\mathbb{R}} \psi^2 \left| \partial_x \Big(
\frac{w}{\psi}\Big) \right|^2 \, dx.
\end{aligned}
\end{equation}
\begin{lemma}[point spectral stability]\label{lemptsp}
Suppose $\tau \geq 0$. If $\lambda \in \sigma_\mathrm{\tiny{pt}} \cap \Omega$ then either $\lambda = 0$, or $\mathrm{Re} \,
\lambda \leq -
\chi_1(\tau) < 0$, for some uniform $\chi_1(\tau) > 0$.
\end{lemma}
\begin{proof}
\smartqed
The result is a consequence of the basic energy estimate \eqref{basicee}. Indeed, suppose that $\lambda \in
\sigma_\mathrm{\tiny{pt}}$ and $\Re \lambda \geq 0$ (and consequently, $\lambda \in \Omega$). Then after the transformation, $w =
e^{-\theta} v \in H^2$ satisfies \eqref{basicee}. Notice that
\[
\Re \int_{\mathbb{R}} \overline{w} w_x \, dx = \tfrac{1}{2} \int_{\mathbb{R}} \partial_x \big( |w|^2\big) \, dx = 0.
\]
First, let us assume that $\tau > 0$. For shortness, we denote,
\[
\begin{aligned}
k_0 &:= (1-c^2 \tau) \int_{\mathbb{R}} \psi^2 \left| \partial_x \Big( \frac{w}{\psi}\Big) \right|^2 \, dx &\geq 0,\\
k_1 &:= (1-c^2 \tau)^{-1} \int_{\mathbb{R}} b(x) |w|^2 \, dx &> 0,\\
k_2 &:= \tau^2 \| w \|_{L^2}^2 &> 0,\\
i k_3 &:= \int_{\mathbb{R}} \overline{w} w_x \, dx,
\end{aligned}
\]
with $k_j \in \mathbb{R}$. Notice that $k_1, k_2 > 0$ because $v$ is an eigenfunction, $\tau > 0$, and because
of \eqref{H2}.
Let us denote $\zeta = \Re \lambda$, $\beta = \Im \lambda$. Therefore, taking the real and imaginary parts of
\eqref{basicee} yields
\[
\begin{aligned}
(\zeta^2 - \beta^2) k_2 + 2c\tau \beta k_3 + \zeta k_1 + k_0 &= 0,\\
2 \zeta \beta k_2 - 2c\tau \zeta k_3 + \beta k_1 &= 0.
\end{aligned}
\]
Multiply the first equation by $\zeta$, the second by $\beta$, and add them up. The result is,
\[
(\zeta^2 + \beta^2) (k_1 + \zeta k_2) + \zeta k_0 = 0,
\]
or, equivalently,
\[
|\lambda|^2 k_1 + (\Re \lambda) \big( k_0 + |\lambda|^2 k_2 \big) = 0.
\]
Since $k_0 > 0$, $k_1, k_2 \geq 0$, this implies that $\Re \lambda \leq 0$.
Now, if we assume that $\zeta = \Re \lambda = 0$, from the equations we have that $\beta^2 k_1 = 0$. Since
$k_1 > 0$ we conclude that $\beta = 0$ and this implies that $\lambda = 0$. On the other hand, if we assume
that $\beta = \Im \lambda = 0$, then from the first equation we obtain,
\[
k_2 \zeta^2 + k_1 \zeta + k_0 = 0,
\]
or,
\[
\zeta = \Re \lambda = - \frac{k_1}{2 k_2} \pm \frac{1}{2 k_2} \Big( k_1^2 - 4 k_2 k_0 \Big)^{1/2}.
\]
Since $k_2 k_0 \geq 0$ we have that $\Re \lambda = \zeta < 0$, a contradiction.
We conclude that the only
eigenvalue with $\Re \lambda = 0$
is $\lambda = 0$ and that, for any other eigenvalue with $\lambda \neq 0$ in $\Omega$, there holds
\[
\Re \lambda \leq - \chi_1(\tau) < 0,
\]
for some $\chi_1(\tau) > 0$. This holds because the set $\sigma_\mathrm{\tiny{pt}}$ comprises isolated eigenvalues with finite multiplicity.
$-\chi_1(\tau) < 0$ is actually the real part of the first (isolated) eigenvalue different from zero. In
other words, there is a spectral gap.
In the case where $\tau = 0$, the basic energy estimate \eqref{basicee} yields
\[
\lambda \int_{\mathbb{R}} b(x) |w|^2 \, dx = - \int_{\mathbb{R}} \psi^2 \left| \partial_x \Big(
\frac{w}{\psi}\Big) \right|^2 \, dx,
\]
which implies, in turn, that $\lambda \in \mathbb{R}$ and $\lambda \leq 0$.
Finally, notice that $\lambda = 0$ if and only if
$w/\psi = 0$ a.e., which is tantamount to $v = U_x$ a.e. This concludes the proof of the lemma.
\qed \end{proof}
As a consequence of the proof of Lemma \ref{lemptsp} we have the following immediate
\begin{corollary}
\label{corgm}
$\lambda = 0$ is an eigenvalue with $g.m. = 1$.
\end{corollary}
\subsection{Simple translation eigenvalue}
We now show that the eigenvalue $\lambda = 0$ is a simple eigenvalue.
\begin{lemma}\label{lemalgm}
The algebraic multiplicity of $\lambda = 0 \in \sigma_\mathrm{\tiny{pt}}$ is equal to one.
\end{lemma}
\begin{proof}
\smartqed
From Corollary \ref{corgm}, we know that $\Phi = (U_x,U_{xx})^\top \in H^1 \times H^1$ is the only
eigenfunction associated to $\lambda = 0$. Let us denote, for simplicity, $\phi = U_x \in H^2$,
so that $\Phi = (\phi, \phi_x)^\top$. Clearly, because of equation \eqref{nprofileq}, $\phi \in H^2$ is a
solution to
\[
{\mathcal{A}} \phi := (1-c^2 \tau) \phi_{xx} + cb(x) \phi_x + a(x) \phi = 0.
\]
This holds upon differentiation \eqref{nprofileq} with respect to $x$. The auxiliary operator, ${\mathcal{A}} : L^2 \to
L^2$ defined above, with domain ${\mathcal{D}}({\mathcal{A}}) = H^2$, has a formal adjoint, ${\mathcal{A}}^* : L^2 \to L^2$, given by
\[
{\mathcal{A}}^* \psi = (1-c^2 \tau) \psi_{xx} - cb(x) \psi_x + (a(x) -cb'(x)) \psi, \qquad \psi \in {\mathcal{D}}({\mathcal{A}}^*) = H^2 \subset
L^2.
\]
Now, for any $\lambda \in \sigma_\mathrm{\tiny{pt}}$, the operator ${\mathcal{T}}^\tau(\lambda)$ is Fredholm with index zero. Therefore, by
properties of closed operators \cite{Kat80}, we have that
\[
\dim \ker {\mathcal{T}}^\tau(\lambda)^* = \dim {\mathcal{R}}({\mathcal{T}}^\tau(\lambda))^\perp = \mathrm{codim} \, {\mathcal{R}}({\mathcal{T}}^\tau(\lambda))
= \dim \ker {\mathcal{T}}^\tau(\lambda).
\]
Since $\dim \ker {\mathcal{T}}^\tau(0) = 1$ we conclude that there exists a unique bounded solution $\Psi =
(y,z)^\top \in H^1 \times H^1$ to the adjoint equation
\[
{\mathcal{T}}^\tau(0)^* \Psi = - \big( \partial_x + \mathbb{A}^\tau(x,0)^*\big)\Psi = 0.
\]
From the expression for $\mathbb{A}^\tau(x,0)$ we observe that $(y,z)^\top \in H^1 \times H^1$ is a solution to the
system
\begin{equation}
\label{yzsyst}
\begin{aligned}
-a(x)z + (1-c^2 \tau) y_x &= 0,\\
(1-c^2 \tau) y - cb(x) z + (1-c^2\tau) z_x &= 0.
\end{aligned}
\end{equation}
Since the coefficents are bounded and $y, z \in H^1$, by a bootstrapping argument we can verify from the
system of equations that actually $y, z \in H^2$. Thus, upon differentiation of the second equation and
substitution into the first one we obtain
\[
{\mathcal{A}}^* z = (1-c^2 \tau) z_{xx} - cb(x) z + (a(x) - b'(x)) z = 0.
\]
We conclude that $z = z(x)$ is the only bounded $H^2$-solution to ${\mathcal{A}}^* z = 0$.
Now, like in \cite{LMPS16}, let us define the Melnikov integral
\[
\Gamma := \langle \Psi, \big( \partial_\lambda \mathbb{A}^\tau(x,\lambda) \big)_{|\lambda = 0} \Phi \rangle_{L^2
\times L^2}.
\]
It is well-known (see section 4.2.1 in \cite{San02}) that $\Gamma$ decides whether $\lambda = 0$ is a simple
eigenvalue: if $\Gamma \neq 0$ then its algebraic multiplicity is equal to one (see also \cite{LMPS16} and the
discussion therein). From \eqref{derivlamA} we observe that $\partial_\lambda \mathbb{A}^\tau(x,\lambda)
\big)_{|\lambda = 0} = \mathbb{A}_1^\tau(x)$, and therefore we arrive at
\[
\begin{aligned}
\Gamma = \langle \Psi, \mathbb{A}_1^\tau(x) \Phi \rangle_{L^2 \times L^2} &= \int_{\mathbb{R}} \begin{pmatrix}
y \\ z
\end{pmatrix}^* \mathbb{A}_1^\tau(x) \begin{pmatrix}
\phi \\ \phi_x
\end{pmatrix} \, dx \\
&= (1-c^2 \tau)^{-1} \int_{\mathbb{R}} \overline{z} \big( b(x) \phi - 2c\tau \phi_x \big) \, dx.
\end{aligned}
\]
Like in the argumentation leading to the proof of Lemma 3.2 in \cite{LMPS16}, a direct computation allows to
verify that the only bounded solution to ${\mathcal{A}}^* z = 0$ is given by $z =\phi/h^2$, where $h$ is a solution to
\[
h_x = - \frac{c b(x)}{2 (1- c^2 \tau)} h,
\]
that is, $h(x) = e^{\theta(x)}$ as in the previous section. By the arguments of Lemma \ref{goodw} it is easy
to verify that $z \in H^2$ inasmuch as $\phi \in H^2$. Thus, a direct computation yields
\[
{\mathcal{A}}^* z = \frac{1}{h^2}\Big( (1-c^2 \tau) \phi_{xx} + cb(x) \phi_x + a(x) \phi \Big) = \frac{1}{h^2} {\mathcal{A}}
\phi = 0,
\]
as claimed. Whence, substituting $z = \phi / h^2$ into the expression for $\Gamma$ we obtain
\[
\begin{aligned}
(1-c^2 \tau) \Gamma &= \int_{\mathbb{R}} \frac{\overline{\phi}}{h^2} \big( b(x) \phi - 2c\tau \phi_x \big) \, dx \\
&= \int_{\mathbb{R}} \frac{b(x)}{h^2} |\phi|^2 - \frac{c\tau}{h^2} \partial_x ( |\phi|^2 ) \, dx \\
&= (1 - c^2 \tau)^{-1} \int_{\mathbb{R}} \frac{b(x)}{h^2} |\phi|^2 \, dx,
\end{aligned}
\]
after integration by parts and substitution of the equation for $h$. We observe that
\[
\Gamma = (1 - c^2 \tau)^{-2} \int_{\mathbb{R}} \frac{b(x)}{h^2} |\phi|^2 \, dx > 0,
\]
and the conclusion follows.
\qed
\end{proof}
\subsection{Main result}
We conclude this section by stating our main theorem.
\begin{theorem}[spectral stability with spectral gap]
\label{mainthm}
Under assumptions \eqref{H1} and \eqref{H2}, for each $\tau \in [0,\tau_m)$ fixed let $U = U(x)$ be the
monotone traveling front solution to \eqref{hypAC}. Then this front is spectrally stable with spectral gap,
more precisely, there exists a uniform $\chi(\tau) > 0$ such that
\[
\sigma \subset \{ \lambda \in \mathbb{C} \, : \, \mathrm{Re} \, \lambda \leq - \chi(\tau) < 0 \} \cup \{ 0 \}.
\]
Moreover, $\lambda = 0$ is a simple isolated eigenvalue (with algebraic multiplicity equal to one) associated
to translation invariance.
\end{theorem}
\begin{proof}
\smartqed
The conclusion follows directly by collecting the results of Corollary \ref{corstabess} and Lemmata \ref{lemptsp}
and \ref{lemalgm}. The spectral gap is given by
\[
\chi(\tau) := \min \{ \chi_0(\tau), \chi_1(\tau) \} > 0,
\]
for each fixed $\tau \in [0, \tau_m)$, where $\chi_0$ is defined in \eqref{defchi0} and $\chi_1$ is the gap defined in Lemma \ref{lemptsp}.
\qed
\end{proof}
Notice that if $\tau > 0$ then the statement of Theorem \ref{mainthm} can be recast in terms of the spectrum of the operators ${\mathcal{L}}^\tau$ defined in \eqref{defcLtau}. Indeed, corollaries \ref{corsamespec} and \ref{corsameptsp} imply that the spectral stability with spectral gap are also properties of the matrix operators ${\mathcal{L}}^\tau$ when $\tau > 0$. Thus, we can state the following
\begin{theorem}
Under assumptions \eqref{H1} and \eqref{H2}, and for each fixed $0 < \tau < \tau_m$ there holds
\[
\sigma({\mathcal{L}}^\tau) \subset \{ \lambda \in \mathbb{C} \, : \, \Re \lambda < - \chi(\tau) < 0\} \cup \{ 0 \},
\]
for some uniform $\chi(\tau) > 0$. Moreover, $\lambda = 0$ is a simple isolated eigenvalue of ${\mathcal{L}}^\tau$ with associated eigenfunction $(U_x, -cU_{xx}) \in \ker {\mathcal{L}}^\tau$.
\end{theorem}
\section{Discussion}
\label{secdisc}
In this paper we established the spectral stability with spectral gap of a family of traveling fronts for nonlinear wave equations of the form \eqref{hypAC} when the reaction function is of bistable type. The equations under consideration are endowed with a positive ``damping" term, $g > 0$, which generalizes the previous studied case of the Allen-Cahn equation with relaxation. To that end, we revisited the existence theory using a dynamical systems approach, more in the spirit of our previous work \cite{LMPS16}. Even though existence results are available in the literature \cite{GiKe15}, here we presented a different construction which allows us to derive a variational formula for the unique wave speed and to establish exponential decay of the profile function. Both features play a role in the stability analysis: the uniqueness of the speed is related to the algebraic multiplicity of the zero eigenvalue of the linearized problem around the front, whereas the exponential decay is crucial to locate the essential spectrum.
Our main result establishes that the spectrum of the linearized problem around the front is located in the complex half plane with negative real part, except for the translation zero eigenvalue, which is isolated with finite multiplicity. This property is also known as \textit{spectral stability with spectral gap} and prevents the accumulation of essential spectrum around zero. In this fashion, we generalize the analysis performed in \cite{LMPS16} for a particular case (the Allen-Cahn equation with relaxation) to a wider class of equations. It is important to remark that this result is more general not only in applicability but also in methodology. Indeed, the present proof makes use of energy estimates and works for the whole parameter regime, whereas the previous argument is of perturbative nature, with an extension to further relaxation times. In our opinion, the method presented here is more direct.
The establishment of spectral stability is a first step in a more general program which includes the nonlinear stability analysis of the fronts under small perturbations. Thanks to the location of the spectrum in the complex plane, we conjecture that the linearized operator around the wave is the infinitesimal generator of a $C_0$-semigroup. The generation of such semigroup and its decaying properties is a matter of future investigation. (As additional information, in the Appendix we present how to establish resolvent estimates in the case of stationary fronts with $c = 0$, yielding the generation of the semigroup via Lumer-Philips theorem.) Such analysis, also called \textit{linearized stability} in the literature \cite{KaPro13,San02}, is used to prove nonlinear stability in a key way. There exist results in the literature which guarantee nonlinear stability under the assumption of spectral stability (see, e.g., Rottmann-Matthes \cite{Rott11,Rott12a}), but they are not applicable to the generic class of equations considered here, as they are restricted to hyperbolic systems with constant coefficient first order operators. We regard the nonlinear stability of the hyperbolic fronts of equations of the form \eqref{hypAC} as an important open problem which warrants attention from the nonlinear wave propagation community.
\begin{acknowledgement}
R. G. Plaza is grateful to the Department of Information Engineering, Computer Science and Mathematics of the
University of L'Aquila, for their hospitality during the Fall of 2017, when this research was carried out.
This work was partially supported by the EU Project ModComShock G.A. N.
642768.
\end{acknowledgement}
\section*{Appendix: Resolvent estimates for stationary fronts}
\addcontentsline{toc}{section}{Appendix}
Fix $\tau > 0$ and consider the space $\mathcal{X} := H^1 \times L^2$ endowed with the scalar product
\begin{equation*}
\langle (u_1,v_1),(u_2,v_2)\rangle_{{}_{\mathcal{X}}}
:=\Re \langle u_1,u_2\rangle_{{}_{L^2}}
+\tau^{-1}\Re \langle \partial_x u_1, \partial_x u_2\rangle_{{}_{L^2}}
+\Re \langle v_1,v_2\rangle_{{}_{L^2}},
\end{equation*}
and corresponding norm
\begin{equation*}
\|(u,v)\|_{{}_{\mathcal{X}}}= \Big( \|u\|_{{}_{L^2}}^2+\tau^{-1}\|u_x \|_{{}_{L^2}}^2 + \|v\|_{{}_{L^2}}^2 \Big)^{1/2}.
\end{equation*}
Then, for simplicity drop the $\tau > 0$ from the notation and consider the operator defined in \eqref{defcLtau},
\[
{\mathcal{L}} \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} c \partial_x & & 1 \\ \tau^{-1} (\partial_x^2 + a(x)) & & \; c \partial_x - \tau^{-1} b(x) \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix},
\]
as a closed, densely defined operator on ${\mathcal{X}}$ with domain ${\mathcal{D}} = H^2 \times H^1$. This operator can be conveniently written as
\[
{\mathcal{L}} = {\mathcal{L}}_0 + {\mathcal{B}},
\]
where
\[
{\mathcal{L}}_0 := \begin{pmatrix} c \partial_x & & 1 \\ \tau^{-1} \partial_x^2 -1 & & c \partial_x - \tau^{-1}b(x) \end{pmatrix}, \qquad {\mathcal{B}} := \begin{pmatrix} 0 & & 0 \\ \tau^{-1} a(x) + 1 & & 0 \end{pmatrix}.
\]
We first observe that the operator ${\mathcal{L}}_0$ is dissipative on $\mathcal{X}$ since for any $\mathbf{w} = (u,v)^\top \in {\mathcal{D}}$,
\begin{equation*}
\begin{aligned}
\langle \mathbf{w}, {\mathcal{L}}_0 \mathbf{w} \rangle_{{}_{\mathcal{X}}} &= \Re \langle u, c u_x + v \rangle_{{}_{L^2}}
+ \tau^{-1} \Re \langle u_x, cu_{xx} + v_x \rangle_{{}_{L^2}} + \\ & \;\;\; + \Re \langle v, \tau^{-1} u_{xx} -u + cv_x - \tau^{-1} b(x) v \rangle_{{}_{L^2}} \\
&= - \tau^{-1} \Re \langle v, b(x)v \rangle_{{}_{L^2}} \leq 0,\\
\end{aligned}
\end{equation*}
in view of Hypothesis \eqref{H2} and having used the fact that $\Re \langle f, f_x\rangle_{{}_{L^2}} = 0$ for any $f \in H^1$. Since ${\mathcal{D}}$ is dense in ${\mathcal{X}}$ and by dissipativity, thanks to the Lumer-Philips theorem (see, e.g., Theorem 12.22 in \cite{ReRo04}) it suffices to show that ${\mathcal{L}}_0 - \lambda$ is onto for real $\lambda$ sufficiently large to conclude that ${\mathcal{L}}_0$ is the infinitesimal generator of a $C_0$-semigroup of contractions, $e^{t {\mathcal{L}}_0}$, satisfying $\| e^{t {\mathcal{L}}_0}\| \leq 1$. Clearly, ${\mathcal{B}}$ is a bounded operator and $\|{\mathcal{B}}\| \leq O(1 + \tau^{-1} \| a \|_{{}_{L^\infty}})$; since ${\mathcal{L}}$ is a bounded perturbation of ${\mathcal{L}}_0$, it is also the infinitesimal generator of a quasi-contractive $C_0$-semigroup, ${\mathcal{S}}(t)$, such that
\[
\| {\mathcal{S}}(t) \| \leq e^{t \| {\mathcal{B}} \|} = e^{tC(1 + \tau^{-1} \| a \|_{{}_{L^\infty}})},
\]
for some $C > 0$ (see Theorem 1.1 in Pazy \cite{Pazy83}, chapter 3).
We illustrate how to prove that ${\mathcal{L}}_0 - \lambda$ is onto for $\lambda$ real and large in the case of a stationary front with $c = 0$ by establishing a resolvent estimate.
First, note that if $c = 0$ then the operator ${\mathcal{L}}_0$ reduces to
\begin{equation}
\label{defopc0}
{\mathcal{L}}_0 \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} 0 & & 1 \\ \tau^{-1} \partial_x^2 - 1 & & \; - \tau^{-1} b(x) \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix}.
\end{equation}
For given $(\phi, \psi)^\top \in {\mathcal{X}}$ suppose that $(u,v)^\top \in {\mathcal{D}}$ is a solution to the resolvent equation
\[
(\lambda - {\mathcal{L}}_0) \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} \phi \\ \psi / \tau \end{pmatrix},
\]
for some $\lambda \in \mathbb{C}$. This yields the system of equations
\begin{equation}\label{reszerospeed}
\lambda u-v=\phi,\qquad
\tau\lambda v-u_{xx} + \tau u+b(x)v=\psi.
\end{equation}
\begin{lemma}\label{lemma:vuprime}
Let $b(x)\geq b_0>0$ for any $x\in\mathbb{R}$.
Given $(\phi,\psi)\in {\mathcal{X}} = H^1 \times L^2$, let $(u,v)\in {\mathcal{D}} = H^2 \times H^1$
be a solution to system \eqref{reszerospeed}.
Then for any $r>0$, there exists a constant $C>0$ (depending on $\tau, b_0$
and $r$) such that
\begin{equation}\label{vuprime}
\|v\|_{{}_{L^2}}+\|u_x\|_{{}_{L^2}}
\leq C\left( \|\psi\|_{{}_{L^2}}+\|\phi_x\|_{{}_{L^2}}+\|u\|_{{}_{L^2}}\right)
\end{equation}
for any $\lambda$ with $|\lambda|\geq r>0$ and $\Re \lambda\geq 0$.
\end{lemma}
\begin{proof}
\smartqed
Multiplying the second equation by $\bar v$ we obtain
\begin{equation*}
\bigl(\tau\lambda+b(x)\bigr) |v|^2-(u_x \bar v)_x + u_x \bar{v}_x + \tau u \bar{v}=\psi \,\bar{v}.
\end{equation*}
Since $v_x = \lambda u_x - \phi_x$, there holds
\begin{equation*}
\bigl(\tau\lambda+b(x)\bigr) |v|^2+\bar \lambda |u_x|^2 + \tau u\,\bar v-(u_x \bar{v}_x)_x
=\psi\,\bar{v}+\bar{\phi}_x \,u_x.
\end{equation*}
Integrating in $\mathbb{R}$ and separating real and imaginary parts, we infer
\begin{equation*}
\begin{aligned}
&\bigl(\tau\Re \lambda+b_0\bigr) \|v\|_{{}_{L^2}}^2+\Re \lambda \|u_x\|_{{}_{L^2}}^2
\leq \|\psi\|_{{}_{L^2}}\|v\|_{{}_{L^2}}+\|\phi_x\|_{{}_{L^2}}\|u_x\|_{{}_{L^2}}
+\tau \|u\|_{{}_{L^2}}\|v\|_{{}_{L^2}},\\
&\bigl|\Im\lambda\bigr|\,\Bigl|\tau\|v\|_{{}_{L^2}}^2-\|u_x\|_{{}_{L^2}}^2\Bigr|
\leq \|\psi\|_{{}_{L^2}}\|v\|_{{}_{L^2}}+\|\phi_x\|_{{}_{L^2}}\|u_x\|_{{}_{L^2}}
+ \tau \|u\|_{{}_{L^2}}\|v\|_{{}_{L^2}}.\\
\end{aligned}
\end{equation*}
Applying Young's inequality, we deduce
\begin{equation*}
\begin{aligned}
&\bigl(\tau \, \Re\lambda+b_0\bigr) \|v\|_{{}_{L^2}}^2+\Re \lambda \|u_x\|_{{}_{L^2}}^2
\leq \frac{1}{b_0}\|\psi\|_{{}_{L^2}}^2+\frac{\tau^2}{b_0}\|u\|_{{}_{L^2}}^2
+\|\phi_x\|_{{}_{L^2}}\|u_x\|_{{}_{L^2}}+\frac{b_0}{2}\|v\|_{{}_{L^2}}^2.
\end{aligned}
\end{equation*}
Hence, the following two estimates hold for any choice of $\lambda$ such that
$\Re\lambda\geq 0$,
\begin{equation}\label{forvuprime}
\begin{aligned}
&\frac12\,b_0\|v\|_{{}_{L^2}}^2+\Re \lambda \|u_x\|_{{}_{L^2}}^2
\leq \frac{1}{b_0}\|\psi\|_{{}_{L^2}}^2+\frac{\tau^2}{b_0}\|u\|_{{}_{L^2}}^2
+\|\phi_x\|_{{}_{L^2}}\|u_x\|_{{}_{L^2}}\\
&\bigl|\Im\lambda\bigr|\,\Bigl|\tau\|v\|_{{}_{L^2}}^2-\|u_x\|_{{}_{L^2}}^2\Bigr|
\leq \|\psi\|_{{}_{L^2}}\|v\|_{{}_{L^2}}+ \tau \|u\|_{{}_{L^2}}\|v\|_{{}_{L^2}}
+\|\phi_x\|_{{}_{L^2}}\|u_x\|_{{}_{L^2}}
\end{aligned}
\end{equation}
For $\Re\lambda\geq c_0>0$, there holds
\begin{equation*}
\begin{aligned}
b_0 \|v\|_{{}_{L^2}}^2+c_0\|u_x\|_{{}_{L^2}}^2
&\leq \frac{1}{b_0}\|\psi\|_{{}_{L^2}}^2+\frac{\tau^2}{b_0}\|u\|_{{}_{L^2}}^2
+\|\phi_x\|_{{}_{L^2}}\|u_x\|_{{}_{L^2}}\\
&\leq \frac{1}{b_0}\|\psi\|_{{}_{L^2}}^2+\frac{1}{2c_0} \|\phi_x\|_{{}_{L^2}}^2
+\frac{\tau^2}{b_0}\|u\|_{{}_{L^2}}^2+\frac{c_0}{2}\|u_x\|_{{}_{L^2}}^2.
\end{aligned}
\end{equation*}
Thus, we deduce
\begin{equation*}
\|v\|_{{}_{L^2}}^2+\|u_x\|_{{}_{L^2}}^2
\leq C\left(\|\psi\|_{{}_{L^2}}^2+\|\phi_x\|_{{}_{L^2}}^2+\|u\|_{{}_{L^2}}^2\right),
\end{equation*}
for some strictly positive constant $C$ depending on $b_0, \tau$ and $c_0$.
Next, let $\lambda$ to be such that $|\Im\lambda|\geq \theta_0>0$.
Then, from the second bound in \eqref{forvuprime}, it follows
\begin{equation*}
\begin{aligned}
\theta_0\|u_x\|_{{}_{L^2}}^2
&\leq \|\psi\|_{{}_{L^2}}\|v\|_{{}_{L^2}}+\|\phi_x\|_{{}_{L^2}}\|u_x\|_{{}_{L^2}}
+\tau\|u\|_{{}_{L^2}}\|v\|_{{}_{L^2}}+\theta_0\tau\|v\|_{{}_{L^2}}^2\\
&\leq \|\psi\|_{{}_{L^2}}\|v\|_{{}_{L^2}}
+\frac{1}{2\theta_0}\|\phi_x\|_{{}_{L^2}}^2+\frac{\theta_0}{2}\|u_x\|_{{}_{L^2}}^2
+\tau \|u\|_{{}_{L^2}}\|v\|_{{}_{L^2}}+\theta_0\tau\|v\|_{{}_{L^2}}^2,
\end{aligned}
\end{equation*}
again thanks to Young's inequality, so that
\begin{equation}\label{uprime}
\|u_x\|_{{}_{L^2}}^2
\leq C\left(\|\psi\|_{{}_{L^2}}^2+\|\phi_x\|_{{}_{L^2}}^2
+\|u\|_{{}_{L^2}}^2+\|v\|_{{}_{L^2}}^2\right),
\end{equation}
for some strictly positive constant depending on $\tau$ and $\theta_0$.
Hence, from the first estimate in \eqref{forvuprime},
we deduce for $\Re\lambda\geq 0$ and $|\Im\lambda|\geq \theta_0>0$, that
\begin{equation*}
\begin{aligned}
\frac12\,b_0\|v\|_{{}_{L^2}}^2
&\leq \frac{1}{b_0}\|\psi\|_{{}_{L^2}}^2+\frac{\tau^2}{b_0}\|u\|_{{}_{L^2}}^2
+\|\phi_x\|_{{}_{L^2}}\|u_x\|_{{}_{L^2}}\\
&\leq \frac{1}{b_0}\|\psi\|_{{}_{L^2}}^2+\frac{\tau^2}{b_0}\|u\|_{{}_{L^2}}^2
+\frac{1}{2\eta}\|\phi_x\|_{{}_{L^2}}^2+\frac{\eta}{2}\|u_x\|_{{}_{L^2}}^2,
\end{aligned}
\end{equation*}
for any $\eta>0$. By choosing $\eta$ sufficiently small and taking advantage of \eqref{uprime},
we deduce
\begin{equation*}
\|v\|_{{}_{L^2}}^2
\leq C\left(\|\psi\|_{{}_{L^2}}^2+\|\phi_x\|_{{}_{L^2}}^2+\|u\|_{{}_{L^2}}^2\right),
\end{equation*}
for some strictly positive constant $C>0$ depending on $\tau, b_0$ and $\theta_0$.
\qed
\end{proof}
Thanks to Lemma \ref{lemma:vuprime}, it is enough to estimate $u$ in $L^2$.
To this aim, we state and prove the following elementary result.
\begin{lemma}
Let $0\leq A\leq B$ with $B>0$.
Given $c_0>0$, set $\Sigma_0:=\{(x,y)\,:\,x\geq 0, |y|\geq c_0\}$. Then
\begin{equation}\label{somesup}
\sup_{(x,y)\in\Sigma_0} \frac{1+A\sqrt{x^2+y^2}}{\bigl(1+B\,x\bigr)|y|}
\leq A+\frac{1}{y_0}.
\end{equation}
\end{lemma}
\begin{proof}
\smartqed
Fix $c_0>0$ and $y\geq c_0$ and consider $y$ such that $|y|\geq y_0$
We want to prove that $M$ is such that
\begin{equation*}
F(x):=M\bigl(1+B\,x\bigr)|y|-A\sqrt{x^2+y^2}\geq 1, \qquad\qquad \forall x\geq 0.
\end{equation*}
Since the function $F$ is concave, it is enough to require that the condition $F(x)\geq 1$
is satisfied at $x=0$ and at $x=+\infty$.
The former condition is satisfied if $M\geq A+1/y_0$;
the latter, if $M>A/By_0$.
Since $B>A$, the first condition implies the second.
\qed
\end{proof}
\begin{lemma}\label{lemma:uell2}
Let $0<b_0\leq b(x)\leq b_1$ for any $x\in\mathbb{R}$.
Given $(\phi,\psi)\in {\mathcal{X}} = H^1 \times L^2$, let $(u,v)\in {\mathcal{D}} = H^2 \times H^1$
be such that \eqref{reszerospeed} holds.
Then there exists $M>0$ such that for any $\theta_0>0$, there exists a constant
$C>0$ (depending on $\tau, b_0, M$ and $\theta_0$) such that
\begin{equation}\label{uell2}
\|u\|_{{}_{L^2}}\leq C\left(\|\phi\|_{{}_{L^2}}+\|\psi\|_{{}_{L^2}}\right),
\end{equation}
for any $\lambda$ with either $\Re\lambda\geq M$ or $|\Im\lambda|\geq \theta_0>0$.
\end{lemma}
\begin{proof}
\smartqed
Multiplying the second equation by $\bar u$ we obtain
\begin{equation*}
\bigl(\tau\lambda^2 +\lambda b(x) + \tau \bigr)|u|^2+|\bar{u}_x|^2-(u_x \bar{u})_x
=(b(x)+\tau\lambda)\bar u\phi+\bar u\psi.
\end{equation*}
Integrating in $\mathbb{R}$ and taking real and imaginary parts, we infer
\begin{equation}\label{realandimag}
\begin{aligned}
\bigl(\tau(\Re\lambda)^2-\tau(\Im\lambda)^2
+b_0\Re\lambda + \tau\bigr)\|u\|_{{}_{L^2}}
&\leq (b_1+\tau|\lambda|)\|\phi\|_{{}_{L^2}}+\|\psi\|_{{}_{L^2}},\\
|\Im\lambda|\bigl(2\tau\Re\lambda+b_0\bigr)\|u\|_{{}_{L^2}}
&\leq (b_1+\tau|\lambda|)\|\phi\|_{{}_{L^2}}+\|\psi\|_{{}_{L^2}},
\end{aligned}
\end{equation}
Applying \eqref{somesup}, from the second inequality in \eqref{realandimag}, we infer
\begin{equation}\label{estimate1}
\|u\|_{{}_{L^2}}
\leq \frac{1}{b_0}\left(\tau+\frac{b_1}{\theta_0}\right)\|\phi\|_{{}_{L^2}}
+\frac{1}{b_0\,\theta_0}\,\|\psi\|_{{}_{L^2}}
\end{equation}
for any $\lambda$ such that $|\Im\lambda|\geq \theta_0>0$ and $\Re\lambda\geq 0$.
Using relations \eqref{realandimag}, we deduce
\begin{equation*}
\begin{aligned}
\bigl(\tau(\Re\lambda)^2 +b_0\Re\lambda + \tau\bigr)\|u\|_{{}_{L^2}}
&\leq (b_1+\tau|\lambda|)\|\phi\|_{{}_{L^2}}+\|\psi\|_{{}_{L^2}}
+\tau(\Im\lambda)^2\|u\|_{{}_{L^2}}\\
&\leq \frac{2\tau\Re\lambda+\tau|\Im\lambda|+b_0}{2\tau\Re\lambda+b_0}
\left((b_1+\tau|\lambda|)\|\phi\|_{{}_{L^2}}+\|\psi\|_{{}_{L^2}}\right).
\end{aligned}
\end{equation*}
For $\Re\lambda$ large and $|\Im\lambda|\leq m_0|\Re\lambda|$, there holds
\begin{equation*}
\|u\|_{{}_{L^2}}\leq \frac{C}{\Re\lambda}\left(\|\phi\|_{{}_{L^2}}
+\frac{1}{\Re\lambda}\|\psi\|_{{}_{L^2}}\right)
\end{equation*}
for some strictly positive constant $C>0$.
\qed
\end{proof}
Collecting the statements contained in Lemma \ref{lemma:vuprime} and Lemma \ref{lemma:uell2},
we deduce the following result.
\begin{proposition}
Given $0<b_0\leq b(x)\leq b_1$ for any $x\in\mathbb{R}$,
let ${\mathcal{L}}_0$ be the operator defined in \eqref{defopc0} on the space ${\mathcal{X}} = H^1 \times L^2$ with
dense domain ${\mathcal{D}} = H^2 \times H^1$. Then,\par
{(i) } there exists $M>0$ such that
\begin{equation*}
\{\lambda \in \mathbb{C} \,:\,\Re\lambda\geq 0\}\setminus[0,M]\subseteq\rho({\mathcal{L}}_0),
\end{equation*}
where $\rho({\mathcal{L}}_0)$ is the resolvent set of ${\mathcal{L}}_0$; and, \par
{(ii) } for any $\theta_0>0$, there exists a constant $C>0$ for which
\[
\|(\lambda-{\mathcal{L}}_0)^{-1}\|\leq C
\]
for any $\lambda$ such that either $\Re\lambda\geq M$ or $|\Im\lambda|\geq \theta_0>0$.
\end{proposition}
\def$'${$'$}
|
{
"timestamp": "2018-02-27T02:01:58",
"yymm": "1802",
"arxiv_id": "1802.08750",
"language": "en",
"url": "https://arxiv.org/abs/1802.08750"
}
|
\section{Preliminaries}\label{Prelims}
Throughout the paper we let $(M, g)$ denote a compact Riemann surface with a smooth metric and let $(X, d)$ denote a compact locally CAT(1) space. We refer the reader to \cite[Section 2.2]{Banff1} for background on CAT(1) spaces. A metric space $(X,d)$ is said to be \emph{locally} CAT(1) if every point of $X$ has a geodesically convex CAT(1) neighborhood. Note that for a compact locally CAT(1) space, there exists a radius $r(X)>0$ such that for all $P \in X$, $\overline {\mathcal B_{r(X)}(P)}$ is a compact CAT(1) space. Let
\[
\tau(X):= \min\{r(X), \pi/4\}
\]and let $\mathrm{inj}(M)$ denote the injectivity radius of $M$.
For $r \in (0, \mathrm{inj}(M))$, $t \in (0, \tau(X))$, we denote geodesic disks and balls in their respective domains as $D_r(x) \subset M$ and $\mathcal B_t(P) \subset X$. We also frequently consider geodesic disks with respect to the metric induced by the pullback of the exponential map and use the same notation, $D_r(0) \subset T_xM =\mathbb R^2$.
Following the definition in \cite{korevaar-schoen1}, the Sobolev space $W^{1,2}(M,X)$ is the space of finite energy maps. That is, $u \in W^{1,2}(M,X)$ if its energy density function (as defined in \cite{korevaar-schoen1}) $|\nabla u|^2 \in L^1(M)$. The total energy of the map $u$ is given by
\[
E[u]:=\int_M|\nabla u|^2 d\mu_g
\]and we denote the energy on subsets $\Omega \subset M$ by
\[
E[u,\Omega]:= \int_\Omega |\nabla u|^2 d\mu_g.
\]Given any $h \in W^{1,2}(\Omega,X)$ we define
\[
W^{1,2}_h(\Omega,X):=\{ f \in W^{1,2}(\Omega,X): Tr(f) = Tr(h)\}
\]where $Tr(u) \in L^2(\partial \Omega,X)$ denotes the trace map (see \cite{korevaar-schoen1}).
\begin{defn}\label{def:min}
A map $u \in W^{1,2}(M,X)$ is \emph{harmonic} if it is locally energy minimizing. In particular, for each $x \in M$ there exist $0<r_x$, $0< \rho< \tau(X)$, and $P \in X$ such that $u(D_{r_x}(x)) \subset \mathcal B_\rho(P)$ and $h:=u|_{D_{r_x}(x)}$ has finite energy and minimizes energy among all maps in $W^{1,2}_h(D_{r_x}(x),\overline{ \mathcal B_\rho(P)})$.
\end{defn} The existence and uniqueness of Dirichlet solutions follows from \cite[Lemma B.2]{Banff2} and \cite{serbinowski}. We will need also the regularity of such solutions.
\begin{theorem}[Lemma 1.3, \cite{Banff1}]
Suppose that $u:D_r \to \mathcal B_{\tau(X)}(P) \subset X$ is an energy minimizing map. Then $u$ is Lipschitz continuous on $D_{r/2}$ with Lipschitz constant depending only on $E[u, D_r]$ and $g$.
\end{theorem}
Let $|u_*(Z)|^2$ denote the directional energy density function for $Z \in \Gamma( TM)$, where $\Gamma(TM)$ is the space of Lipschitz vector fields on $M$ (see \cite[Section 1.8]{korevaar-schoen1}). For any finite energy map $u:(M,g) \to (X,d)$, let
\[
\pi:\Gamma(TM) \times \Gamma(TM) \to L^1(M)
\]where
\[
\pi(Z,W):= \frac 14 \left|u_*(Z+W)\right|^2 - \frac 14\left|u_*(Z-W)\right|^2.
\]By \cite[Lemma 3.5]{Banff1}, $\pi$ is a continuous, symmetric, bilinear, non-negative tensorial operator.
Let
\begin{equation*}
\Phi_u = \pi \left( {\partial_x}, {\partial_x} \right) - \pi\left({\partial_y},{\partial_y} \right) - 2 \mathbf{\textit{i}}\pi \left({\partial_x} , {\partial_y}\right)
\end{equation*}
denote the \emph{Hopf function} for $u$. As in the smooth setting, when $u$ is harmonic, $\Phi_u$ is holomorphic (see \cite[Lemma 3.7]{Banff2}).
\section{Analogues of Classical Results} \label{Tools}
In the smooth setting, the compactness follows from four properties of harmonic maps (see \cite[Proposition 1.1]{Parker}). We state an analogous proposition for harmonic maps into compact locally CAT(1) spaces. Note that the uniform convergence statement is not as strong as Parker's; we can get only $C^0$ uniform convergence. Nevertheless, we are still able to prove Theorem \ref{MAIN}.
\begin{prop}\label{BTtools}
There exist positive constants $C', \epsilon'>0$ depending only on $(M, g)$ and $(X,d)$ such that the following hold:
\begin{enumerate}
\item (Sup Estimate) Let $u:D_r\to X$ be a harmonic map with $E\left[ u,D_r \right]< \epsilon'$ and $0<r<\epsilon'$. Then
\[
\max_{0 \le \sigma \le r} \sigma^2 \sup_{D_{r - \sigma}} |\nabla u|^2 \le C'.
\]In particular for all $x \in D_{3r/4}$,
\[
|\nabla u|^2(x) \leq \frac {C'}{r^2}.
\]
\item (Energy Gap) If $(M,g)=(\mathbb S^2, g_0)$, where $g_0$ is the standard metric on the sphere, and $E\left[ u,\mathbb S^2 \right] < \epsilon'$, then $u$ is a constant map.
\item (Uniform Convergence) Let $u_k:D_r \to X$ be a sequence of harmonic maps with $E\left[ u_k,D_r \right]<\epsilon'$. Then a subsequence $u_k$ convergence in $C^0$ uniformly to a harmonic map $u$ on $D_{r/2}$.
\item (Removable Singularity) Let $u:D_r \backslash \{0\} \to X$ be a finite energy harmonic map. Then $u$ extends to a locally Lipschitz harmonic map $u:D_r \to X$.
\end{enumerate}
\end{prop}
The entirety of this section is devoted to proving each of these results. The results are listed in the order in which they are proven and each subsection contains the proof of a single item.
In the smooth setting, the proofs of these results rely on the Euler-Lagrange equation of the (perturbed) energy functional. Lacking such an equation, we instead exploit weak differential inequalities which follow from the locally minimizing property of harmonic maps coupled with the local convexity of the target space.
\subsection{Sup Estimate}
Following the now classical methods of \cite{Choi-Schoen}, we use a monotonicity formula and scale invariance to prove pointwise gradient bound for harmonic maps with small energy.
\begin{prop}\label{GradientProp}
Suppose $u : D_r \to X$, $r \leq 1$, is a finite energy harmonic map. There exists an $\epsilon_0 >0$, depending only on the metric $g$, such that if $E\left[ u , D_r \right] < \epsilon_0$ and $r < \epsilon_0$, then
\begin{equation}
\max_{0 \le \sigma \le r} \sigma^2 \sup_{D_{r - \sigma}} |\nabla u|^2 \le C_0^2 ,
\end{equation} where $C_0$ depends only on the metric $g$.
\end{prop}
Before proceeding with the proof, we point out an important subharmonicity estimate that we will need. The result follows from a local Bochner type inequality (see \cite{FZ}).
\begin{prop}\label{SubharmProp}
Let $u : (D_{2r},g) \to X$ be a harmonic map with finite energy and let $g$ be a metric with bounded curvature. Then for all $\eta \in C_0^\infty(D_{r})$,
\begin{equation}
-\int_{D_r} \nabla |\nabla u|^2 \cdot \nabla \eta \ge -C' \int_{D_r} \eta |\nabla u|^2 \left( 1+|\nabla u|^2\right)
\end{equation}where $C'>0$ depends only on the curvature of the domain.
\end{prop}
\begin{proof}
For each $x \in \overline {D_r}$, let $s_x:= \sup\{s>0: u(D_s(x)) \subset \mathcal B_{\tau(X)}(u(x))\}$. For the open cover $\{D_{s_x}(x)\}_{x \in \overline{D_r}}$, consider a finite subcover $\left\{D_{s_i}(x_i)\right\}_{1\le i \le m}$ and denote $s:= \min_i\{s_{i}\}$. By \cite {FZ} there exists $C'>0$ depending on the curvature of $M$ such that for each $x_i$,
\begin{equation}
-\int_{D_s(x_i)} \nabla |\nabla u|^2 \cdot \nabla \eta \ge -C' \int_{D_s(x_i)} \eta |\nabla u|^2 \left( 1+|\nabla u|^2\right).
\end{equation}
Now let $\{\phi_i\}$ be a smooth partition of unity subordinate to the covering. Then $\phi_i \in C^\infty_0 (D_s(x_i))$ for each $i$. Moreover, $\sum_i \phi_i \equiv 1$, $\sum_i \nabla \phi_i \equiv 0$. Therefore for any test function $\eta \in C^1_0 (D_r)$,
\begin{eqnarray*}
-\int_{D_r} \nabla |\nabla u|^2 \cdot \nabla \eta &=&- \int_{D_r} \nabla |\nabla u|^2 \cdot \nabla \left(\eta \left(\sum_i \phi _i\right)\right) \\ &=& -\sum_i \int_{D_s(x_i)\cap D_r} \nabla |\nabla u|^2 \cdot \nabla (\eta \phi_i ) \\ &\ge& -C' \sum_i \int_{D_s(x_i)\cap D_r} \eta \phi_i |\nabla u|^2 \left( 1+|\nabla u|^2\right) \\
&=& -C' \int_{D_r} \; \eta|\nabla u|^2 \left( 1+|\nabla u|^2\right).
\end{eqnarray*}
\end{proof}
\begin{proof}[Proof of Proposition \ref{GradientProp}]
Choose $\sigma_0 \in (0 , r] $ and $x_0 \in \overline D_{r - \sigma_0}$ so that
\begin{equation*}
\sigma_0^2 \sup_{D_{r - \sigma_0}} |\nabla u|^2 = \max_{\sigma \in (0,r]} \sigma^2 \sup_{D_{r - \sigma}} |\nabla u|^2,
\end{equation*}
and
\begin{equation*}
|\nabla u|^2 (x_0) \geq \frac 12 \sup_{D_{r - \sigma_0}} |\nabla u|^2.
\end{equation*}
We deduce that
\begin{equation*}
\sup_{D_{\frac{\sigma_0}{2}}(x_0)} |\nabla u|^2 \le 8 |\nabla u|^2 (x_0).
\end{equation*}
Notice that if $\sigma_0^2 |\nabla u|^2 (x_0) \le 4$ then the desired result holds. So suppose instead that $ |\nabla u|^2 (x_0) \ge 4 \sigma_0^{-2}$. Let $\tilde{u}: D_1 \to X $ be given by
\begin{equation*}
\tilde{u}(x) = u \left( x_0 +|\nabla u|^{-1} (x_0) x \right)
\end{equation*}
Then
\begin{equation*}
\sup_{D_1} \left| \nabla \tilde{u} \right|^2 \le 8 \quad and \quad \left| \nabla \tilde{u} \right|^2(0) = 1.
\end{equation*}
By Lemma \ref{SubharmProp}, for all $\eta \in C_0^\infty(D_1)$,
\begin{equation*}
-\int_{D_1} \nabla \left| \nabla \tilde{u} \right|^2 \cdot \nabla \eta \ge -C'\int_{D_1}\eta |\nabla \tilde u|^2 \left( 1+|\nabla \tilde u|^2\right) \geq -9C' \int_{D_1}\eta \left|\nabla \tilde u\right|^2.
\end{equation*} Finally, Morrey's mean value inequality and the scale invariance of the energy implies that, for $c$ depending only on the domain metric $g$,
\begin{equation*}
1 = \left| \nabla \tilde{u} \right|^2 (0) \le c \int_{D_1} \left| \nabla \tilde{u} \right|^2 \leq c \epsilon.
\end{equation*}For $\epsilon$ sufficiently small, we get a contradiction.
\end{proof}
\subsection{Energy Gap}
\begin{prop}[Energy Gap]\label{GapThm}There exists $\epsilon_{\mathrm{gap}}>0$ depending only on $g_0, (X,d)$ (where $g_0$ is the standard metric on $\mathbb S^2$) such that the following holds:
Let $u:\mathbb S^2 \to X$ be a conformal, harmonic map such that $E\left[ u,\mathbb S^2 \right] < \epsilon_{\mathrm{gap}}$. Then $u$ is a constant map.
\end{prop}
\begin{proof}
Suppose first that $u(\mathbb S^2) \subset \mathcal B_{\tau(X)} (P)$ for some $P \in X$. Then, by \cite[Lemma 4.3]{Banff1}, $\Delta d^2(u(x),P) \geq \frac 12 |\nabla u|^2 \geq 0$ holds weakly on all of $\mathbb S^2$. It follows that $d^2(u(x),P) \equiv 0$, i.e. $u$ is constant.
Now suppose that $u(\mathbb S^2)$ is not contained in $\mathcal B_{\tau(X)}(P)$ for all $P \in X$. Then, $\mathrm{diam}(u(\mathbb S^2)) > \tau(X)$. By the Monotonicity Formula of \cite[Theorem 3.4]{Banff2}, there exists $C>0$, independent of $u$ and $p \in \mathbb S^2$ such that
\[
E\left[ u,\mathbb S^2 \right] \geq E\left[ u, u^{-1}(\mathcal B_{\tau(X)}(u(p))) \right] \geq C\tau(X)^2.
\]
Thus, choosing $\epsilon_{\mathrm{gap}} <C \tau(X)^2$ implies the result.
\end{proof}
\subsection{Uniform Convergence}
\begin{prop}There exists $\epsilon_2>0$, depending only on $g, (X,d)$ such that the following holds:
Let $u_k:D_r \to X$ be a sequence of harmonic maps with $E\left[ u_k,D_r \right]<\epsilon_2$. Then a subsequence $u_k$ converges in $C^0$ uniformly to a harmonic map $u$ on $D_{r/2}$.
\end{prop}
\begin{proof}Let $\epsilon_0, C_0$ be as in Proposition \ref{GradientProp}. Set $0<\epsilon_2 \leq \epsilon_0$. Then for all $x,y \in D_{3r/4}$ with $d_g(x,y)<\frac{\tau(X)}{C_0}r$ and all $k$, $d(u_k(x), u_k(y)) < \tau(X)$. Set $s:= \min\left\{\frac{\tau(X)}{C_0}r, \frac r{16}\right\}$. Cover $D_{r/2}$ by disks $\{D_{s/4}(x_j)\}$. By \cite[Remark 3.2]{Banff2}, for all $k$, $u_k|_{D_{s}(x_i)}$ is energy minimizing. By \cite[Theorem 1.3]{Banff1}, the $u_k$ are equicontinuous on the cover $\{D_{s/2}(x_j)\}$. Therefore a subsequence $u_k \to u$ uniformly on every ball in this cover and thus on $D_{r/2}$. Applying \cite[Theorem 2.3]{Banff2} to each disk $D_{s/2}(x_j)$, we see that $u$ is energy minimizing on each disk $D_{s/4}(x_j)$. It follows that $u$ is harmonic on $D_{r/2}$.
\end{proof}
\subsection{Removable singularity theorem}
Notice that the work of this subsection extends the result of \cite[Theorem 3.6]{Banff2}, where a removable singularity theorem is proven for conformal harmonic maps.
\begin{theorem}\label{RSThm}
Let $u:D_1 \backslash \{0\} \to X$ be a finite energy harmonic map. Then $u$ extends to a locally Lipschitz harmonic map $u:D_1 \to X$.
\end{theorem}
\begin{proof}
Since $u$ has finite energy, the Hopf function $\Phi_u\in L^1(D_1 \backslash \{0\}, \mathbb C \backslash \{0\})$ and therefore $\Phi_u$ can have at worst a simple pole at the origin. Without loss of generality, assume that $\Phi_u$ is nowhere zero on $D_1\backslash \{0\}$.
We now follow the ideas of Schoen \cite[Theorem 10.4]{Schoen-Analytic} to define a conformal harmonic map. Schoen's argument involves taking the square root of $-\Phi_u$, and since in his case the domain is a disk and the image does not contain the origin the square root function is well-defined. We build an admissible cell complex $W$ (see \cite[Section 2.1]{Banff1}, \cite[Section 2.2]{daskal-meseCAG}) such that $W \setminus W^{(0)}$ will be the double cover of $\mathbb C \setminus \{0\}$. We then lift the map $\Phi_u$ to be defined from this double cover, allowing us to take its square root.
Let $H_j:=\{z \in \mathbb C: \im(z) \geq 0\}$, $j=1, \dots, 4$ denote four $2$-cells and let $z_j = x_j+ i y_j$ denote the coordinates in the $2$-cell $H_j$. Define the $2$-complex $W:= \bigsqcup_{j=1}^4H_j / \sim$ where the similarity relations determine the gluing of $1$-cell boundaries and are non-empty relations only in the following cases:
\[
\left\{ \begin{array}{ll}
z_1 \sim z_2 &\text{ iff } \re(z_1)=\re(z_2) \leq 0, \im(z_1)=\im(z_2)=0,\\
z_2\sim z_3 &\text{ iff } \re(z_2)=\re(z_3) \geq 0, \im(z_2)=\im(z_3)=0,\\
z_3 \sim z_4 &\text{ iff } \re(z_3)=\re(z_4) \leq 0,\im(z_3)=\im(z_4)=0,\\
z_4 \sim z_1 &\text{ iff } \re(z_4)=\re(z_1) \geq 0,\im(z_4)=\im(z_1)=0.
\end{array}\right.
\]It is straightforward to see that $W\backslash W^{(0)}$ is a double cover of $\mathbb C\backslash \{0\}$. We will associate each $p \in W$ with a projection onto $\mathbb C$ using isometries of half-spaces.
Let $\psi_j:H_j \to \{z \in \mathbb C: \im(z) \geq 0\}$, $ \psi_j^-:H_j \to \{z \in \mathbb C: \im(\overline z) \geq 0\}$ denote the natural Euclidean isometries. For $p \in W$, we define $\re, \im:W \to \mathbb R$ such that
\[
\re(p):=
\re(\psi_j(z_j)) \text{ if } p=z_j, j=1,\dots, 4,
\]
\[
\im(p):=\left\{ \begin{array}{ll}
\im(\psi_j(z_j)) &\text{ if } p=z_j, j=1,3,\\
\im(\psi_j^-(z_j)) & \text{ if } p = z_j, j=2,4.
\end{array}\right.
\]Let $\Pi: W \to \mathbb C$ such that
$\Pi(p):= \re(p) + i \im(p)$.
We define $\underline u: W \backslash W^{(0)} \to X$ and $\underline \Phi_u: W \backslash W^{(0)} \to \mathbb C \backslash \{0\}$ such that
\[
\underline u(p) := u \circ \Pi(p); \quad \quad \underline \Phi_u(p):= \Phi_u \circ \Pi (p).
\]Note that $(\Phi_u)_*(\pi_1(D_1 \backslash \{0\})) = n \mathbb Z$ for some $n \in \mathbb N$. It follows that $(\underline \Phi_u)_*(\pi_1(W \backslash W^{(0)}))=2n\mathbb Z \subset 2\mathbb Z$. Therefore, there exists a map $\Psi_u:W \backslash W^{(0)}\to \mathbb C \backslash \{0\}$ such that $\Psi_u^2(p) = \underline \Phi_u(p)$.
Define $v:W \backslash W^{(0)} \to \mathbb R$ such that
\[
v(p):= \re \int_{p_0}^p\Psi_u(\zeta) d\zeta
\]where $p_0 \in W \backslash W^{(0)}$. By construction, $v$ is a well-defined, real-valued harmonic function which is minimizing on every compact subset of $W \backslash W^{(0)}$.
We compute
\begin{equation*}
\frac{\partial v}{ \partial z}(p) = \frac{1}{2} \re \Psi(p) \in \mathbb R.
\end{equation*} It follows that $E[v] \leq C\int_{D_1\backslash \{0\}} |\Phi_u| d\mu_g< \infty$ and thus $v$ has finite energy. Let $\tilde{u} :W \backslash W^{(0)}\to X \times \mathbb R$ where
\[
\tilde{u}(p) := \left(\underline u(p) , v(p) \right).
\] By definition, $\tilde u$ is a finite energy harmonic map and
the Hopf differential of $\tilde{u}$ satisfies
\begin{equation*}
{\Phi_{\tilde u}} (p) = \Phi_u(p) + 4 \left(\frac{\partial v}{\partial z} \right)^2 (p) \equiv 0.
\end{equation*}
Therefore, $\tilde{u} :W \backslash W^{(0)} \to X \times \mathbb R$ is a \emph{conformal} harmonic map. We apply \cite[Theorem 3.6]{Banff2} to prove the removable singularity result for $\tilde u$. Observe that the hypothesis of the cited theorem states that the target space is \emph{compact} locally CAT(1) and that the domain is a Riemann surface. Nevertheless, the theorem can still be applied.
While the metric space $(X \times \mathbb R, d \times \delta)$ is obviously not compact, it remains a locally CAT(1) space. Moreover, for each $P \in X \times \mathbb R$, the closed geodesic ball $\overline{\mathcal B_{\tau(X)}(P)}\subset X \times \mathbb R$ is a compact locally CAT(1) space. It follows that for any $\rho \in (0, \tau(X))$ and any $y \in X \times \mathbb R$, $\tilde u$ is energy minimizing on the domain $\tilde u^{-1}(\mathcal B_{\rho}(P))$. The removable singularity theorem for conformal harmonic maps does not in fact require compactness of the target space but does require a uniform lower bound in the target for which harmonic maps are minimizers. (This uniform lower bound is needed in order to appeal to the monotonicity formula.) Our target possesses such a uniform lower bound.
Moreover, while our domain here is a cell complex, away from $W^{(0)}$ the complex is a Riemannian manifold with a smooth Riemannian metric.
Therefore everywhere we apply the arguments of \cite[Theorems 3.4 and 3.6]{Banff2} the fact that the domain is a complex is irrelevant. It follows that $\tilde u$ extends as a locally Lipschitz harmonic map $\tilde u:W \to X \times \mathbb R$ and thus so does $u$.
\end{proof}
\section{Isoperimetric Inequality for Minimal Surfaces with Small Area} \label{IsoperimetricSection}
We prove an isoperimetric inequality for minimal surfaces with small area in a CAT(1) metric space. By a minimal surface we mean a conformal harmonic map $u : \left( \Sigma , g \right) \to X$ which is minimizing in the sense of Definition \ref{def:min}. For such a map $u$, we define the area of its image by integrating the conformal factor $\lambda= \frac 12|\nabla u|^2$:
\begin{equation*}
\mathrm{Area} \left( \mathrm{image}(u) \right) = \int_{\Sigma}\; \lambda \; d\mu_g.
\end{equation*}
To prove the isoperimetric inequality we follow the classical arguments of Hoffman-Spruck \cite{HoffmanSpruck} who prove the result by first proving a Sobolev inequality for $C^1$ functions.
We begin by improving the weak differential inequality satisfied by $d^2(u(x),Q)$ for some fixed $Q\in X$.
\begin{lemma}\label{lem:2.5-modified}
Given a geodesic triangle $\triangle PQS\subset X$ and $0 \leq \eta, \eta' \leq 1$, let $P_{\eta'}:= (1-\eta')P + \eta'Q$ and $S_\eta:=(1-\eta)S + \eta Q$. Then\begin{align*}
d^2 \left( P_{\eta'} , S_\eta \right) &\le \left( 1 - 2 \eta d_{QS} \cot d_{QS} \right) d^2_{PS} - 2 \left( \eta - \eta' \right) \left( d_{QS} - d_{QP} \right) d_{QS} + (\eta' - \eta)^2 d_{QS}^2 \\
& \quad+\mathrm{Quad}(\eta, \eta') \mathrm{Quad}(d_{PS}, d_{QS}-d_{QP})+ \mathrm{Cub}\left( d_{PS}, d_{QS}-d_{QP}, \eta-\eta' \right).
\end{align*}
\end{lemma}
\begin{proof}
The proof follows from \cite[Lemmas 2.4 and 2.5]{Banff1} by keeping and expanding the equality
\[
\frac{\sin^2((1-\eta) d_{QS})}{\sin^2 d_{QS}} = \left(1 - \eta \frac {d_{QS}}{\sin d_{QS}} \cos d_{QS} +O(\eta^2)\right)^2
\]rather than getting an upper estimate.
\end{proof}
We now prove a modification of \cite[Lemma 4.3]{Banff1}, which implies almost subharmonicity for $d(Q,u(x))$.
\begin{lemma}\label{lem:div-thm}
Let $0 < t < \tau(X)$ and $u : \left( D_r , g \right) \to \mathcal{B}_t (P) \subset X$ be an energy minimizing map. For a fixed $Q \in \mathcal{B}_t (P) $, $\eta \in [0,1]$, and all $0 < \sigma \le r$,
\begin{equation*}
\int_{D_\sigma} \; 2 \eta \; \hat{d} \; \cot \hat{d} \; \left| \nabla u \right|^2 d\mu_g \le -\int_{D_\sigma} \; \left< \nabla \eta , \nabla \hat{d}^2 \right> d\mu_g ;
\end{equation*}
where $\hat{d} (x) := d \left( Q , u(x) \right)$.
\end{lemma}
\begin{proof}
Define
$u_{\eta}:(D_\sigma,g) \rightarrow X$
by setting
\[
u_{\eta}(x)=(1-\eta(x)) u(x)+\eta(x) Q
\]
for $\eta \in C^{\infty}_c (D_\sigma)$.
Letting $S = u(x),P = u(y), \eta' =\eta(y)$, we use the estimate of Lemma \ref{lem:2.5-modified} to observe that for $\hat d(x):= d(Q, u(x))$,
\begin{align*}
d^2(u_\eta(y), u_\eta(x)) &\leq (1- 2\eta(x)\hat d(x)\cot(\hat d(x)))d^2(u(x),u(y)) \\& \quad
-2(\eta(x)-\eta(y))(\hat d(x)-\hat d(y))\hat d(x)\\
& \quad +(\eta(y)-\eta(x))^2\hat d^2(x) + \eta^2(x)\mathrm{Quad}(d(u(x),u(y)), \hat d(x)-\hat d(y)) \\
&\quad + \mathrm{Cub}\left( d(u(x),u(y)), \hat d(x)-\hat d(y), \eta(x)-\eta(y) \right).
\end{align*} The rest of the proof is identical to the rest of the proof of \cite[Lemma 4.3]{Banff1}.
\end{proof}
\begin{lemma}
Let $u: \Sigma\to X$ be a conformal harmonic map. Suppose $\xi \in C^1 \left( -\infty , \infty \right)$ is a non-decreasing function such that $\xi(t) = 0$ for $t\le 0$, $h \in C_0^1(\Sigma)$ is a non-negative function, and $\xi h \in [0,1]$. For $x_0 \in \Sigma$ and $0<\rho<\tau(X)$, define
\begin{equation*}
\phi_{x_0} (\rho) := \int_\Sigma \; h(x) \; \xi (\rho - r(x)) \; \lambda(x) d\mu_g;
\end{equation*}
and
\begin{equation*}
\psi_{x_0}(\rho) := \int_\Sigma \; \left| \nabla h \right|(x) \; \xi (\rho - r(x)) \; \lambda^{\frac{1}{2}}(x) d\mu_g
\end{equation*}
where $r(x) := d\left( u(x) , u(x_0) \right)$. Then the following differential inequality holds weakly:
\begin{equation}\label{eq:iso-1}
- \frac{d}{d \rho} \left( \frac{\phi_{x_0}(\rho)}{\sin^2 \rho} \right) \le \frac{\psi_{x_0}(\rho)}{\sin^2\rho}.
\end{equation}
\end{lemma}
\begin{proof}
First note that \eqref{eq:iso-1} is equivalent to
\begin{equation}\label{form2}
2 \cot \rho\, \phi_{x_0}(\rho) \le \psi_{x_0}(\rho) + \phi'_{x_0}(\rho).
\end{equation} By Lemma~\ref{lem:div-thm}, for any test function $\Psi\in [0,1]$ and $x_0 \in \Sigma$, we have that
\begin{equation*}
\int_{\Omega} \; 2 \Psi r \cot r \left| \nabla u \right|^2 d\mu_g \le - \int_{\Omega} \; \left< \nabla \Psi , \nabla r^2 \right> d\mu_g
\end{equation*}
where $\Omega:= u^{-1} \left( \mathcal B_\rho (u(x_0)) \right) $.
Let $\Psi (x) = h(x) \xi \left( \rho - r(x) \right)$ so that
\begin{equation*}
\nabla \Psi (x) = -h(x) \xi'\left( \rho - r(x) \right) \nabla r(x) + \xi \left( \rho - r(x) \right) \nabla h(x).
\end{equation*}
By conformality and given the support of $\xi, \xi'$, it follows that
\begin{eqnarray}
2 \rho \cot \rho \; \int_{\Sigma} \; \Psi \; \lambda d\mu_g &\le& \int_{\Sigma} \; \Psi r \cot r \; |\nabla u|^2 d\mu_g \notag \\ &\le& \int_{\Sigma} \; r(x) h(x) \xi'\left( \rho - r(x) \right) \left|\nabla r(x) \right|^2 \; d\mu_g \notag \\ && - \int_{\Sigma} \; r(x) \xi \left( \rho - r(x) \right)\langle \nabla h(x) , \nabla r(x) \rangle\; d\mu_g \notag \\ &\le& \notag \int_{\Sigma} \; r(x) h(x) \xi'\left( \rho - r(x) \right) \; \lambda d\mu_g \notag \\ && + \int_{\Sigma} \; r(x) \xi \left( \rho - r(x) \right) \left| \nabla h(x) \right| \lambda^{\frac{1}{2}} \; d\mu_g \notag \\ &\le& \rho \int_{\Sigma} \; h(x) \xi'\left( \rho - r(x) \right) \; \lambda d\mu_g \notag \\ && + \rho \int_{\Sigma} \; \xi \left( \rho - r(x) \right) \left| \nabla h(x) \right| \lambda^{\frac{1}{2}} \; d\mu_g. \notag
\end{eqnarray}
Note that the string of inequalities implies that \eqref{form2} holds weakly.
\end{proof}
\begin{lemma}
Let $u: \left(\Sigma , g \right) \to X$ be a minimal surface. Let $x_0 \in \Sigma$ with $h(x_0) \ge 1$. Let $\alpha$ and $t$ satisfy $0< \alpha < 1 \le t$. Set
\begin{equation*}
\rho_0 := \sin^{-1} \left( \frac{\int_\Sigma h(x) \; \lambda (x) \; d \mu_g}{\pi (1 - \alpha)} \right)^{\frac 12} ,
\end{equation*}
\begin{equation*}
\overline{\phi}_{x_0} (\rho) := \int_{S_\rho (x_0)} h(x) \lambda(x) d\mu_g ,
\end{equation*}
and
\begin{equation*}
\overline{\psi}_{x_0}(\rho) := \int_{S_\rho (x_0)} \left| \nabla h(x) \right| \lambda^{\frac{1}{2}}(x) d\mu_g
\end{equation*} where
\[
S_\rho(x_0):=\{ x \in \Sigma: d(u(x), u(x_0))< \rho\}.
\]
Then there exist $\rho$ with $0 < \rho < \rho_0$ such that
\begin{equation*}
\overline{\phi}_{x_0} (t \rho) \le \alpha^{-1} \rho_0 \overline{\psi}_{x_0}(\rho);
\end{equation*}
provided that
\begin{equation*}
\frac{\int_\Sigma h(x) \; \lambda (x) \; d \mu_g}{\pi (1 - \alpha)} \le 1
\end{equation*}
and
\begin{equation*}
t \rho_0 \le \tau(X).
\end{equation*}
\end{lemma}
\begin{proof}
The proof follows exactly the outline of \cite[Lemma 4.2] {HoffmanSpruck}, taking advantage of the differential inequality in \eqref{eq:iso-1} to establish a contradiction.
\end{proof}
An argument similar to the covering argument used in \cite[Theorem 2.1]{HoffmanSpruck} (see also \cite{MichaelSimon}) immediately implies the following lemma.
\begin{lemma}\label{lem:sob-type}
Let $u: \left( \Sigma,g \right) \to X$ be a conformal harmonic map with $\mathrm{image}(u) \subset \mathcal B_{\tau(X)} (P)$. If $ \mathrm{Area}\left[ u \left( \Sigma \right) \right] \le \frac{\pi}{3} $,
then for any $h \in C^1(\Sigma)$,
\begin{equation*}
\left( \int_\Sigma \; h^2(x) \; \lambda(x) \; d \mu_g \right)^{\frac{1}{2}} \le \left( \frac{27 \pi}{4} \right)^{\frac{1}{2}} \int_\Sigma \; \left| \nabla h \right|(x) \; \lambda^{\frac{1}{2}}\; d \mu_g.
\end{equation*}
\end{lemma}
Using the Sobolev type inequality of Lemma \ref{lem:sob-type} and an argument adapted from \cite{mese-iso}, we prove the isoperimetric inequality.
\begin{theorem}\label{thm:isoperimetric}
Let $u: \left( \Sigma ,g \right) \to X$ be a conformal harmonic map with $\mathrm{image}(u) \subset \mathcal B_{\tau(X)} (P)$. If $ \mathrm{Area}\left[ u \left( \Sigma \right) \right] \le \frac{\pi}{3} $,
then
\begin{equation*}
\frac{1}{2} E(u) = \mathrm{Area} \left[ u\left( \Sigma \right) \right] \le \left( \frac{27 \pi}{4} \right) \mathrm{length}^2 \left[ u\left( \partial \Sigma \right) \right],
\end{equation*}
\end{theorem}
\begin{proof}
Since $u$ is uniformly continuous, for any $\epsilon>0$, we can pick a family of Lipschitz closed curves $\Gamma_\epsilon$ that approximate $\partial \Sigma$ i.e. with
\begin{equation*}
\left| \mathrm{length} \left[ u(\Gamma_\epsilon) \right] - \mathrm{length} \left[ u\left( \partial \Sigma \right) \right] \right| < \left(\frac 4{27\pi}\right)^{\frac 12} \epsilon.
\end{equation*}
and such that
\begin{equation*}
\mathrm{Area}^{\frac 12}\left[ u\left( \Sigma \right) \right] < \epsilon + \mathrm{Area}^{\frac 12}\left[ u\left( \Sigma_\epsilon \right) \right]
\end{equation*}
where $\Sigma_\epsilon$ is the connected component of $\Sigma \backslash \Gamma_\epsilon$ which is disjoint from $\partial \Sigma$. By \cite[(1.9xvi)]{korevaar-schoen1}, for any Lipschitz closed curve $\Gamma \subset \Sigma$,
\[
\mathrm{length}[u(\Gamma)] = \int_\Gamma \lambda^{\frac 12} d\sigma_\Gamma.
\]
Following the proof of \cite[Theorem 6.1]{meseCAG}, let $\lambda^\sigma:=e^{(\log \lambda)_\sigma}$, where $(\log \lambda)_\sigma$ is a symmetric mollification of $\log \lambda$. Then, $\lambda^\sigma \ge \lambda$ and $\sqrt{\lambda^\sigma} \to \sqrt{\lambda}$ in $W^{1,2}_{loc}(\Sigma)$. Then for any $h \in C^\infty_0(\Sigma)$ with $\|h\|_{L^\infty}<1$ and $\sigma>0$ sufficiently small,
\[
\int_\Sigma |\nabla h| \lambda^{\frac 12} d\mu_g \leq \int_\Sigma |\nabla h| (\lambda^\sigma)^{\frac 12}d\mu_g,
\]
\[
\int_\Sigma h^2 \lambda^\sigma d\mu_g \leq \int_\Sigma h^2 \lambda d\mu_g + \int_\Sigma(\lambda^\sigma - \lambda) d\mu_g.
\]By Lemma \ref{lem:sob-type},
\[
\left(\int_\Sigma h^2 \lambda^\sigma d\mu_g \right)^{\frac 12} \leq \left(\frac{27\pi}4\right)^{\frac 12} \int_\Sigma |\nabla h|(\lambda^\sigma)^{\frac 12}d\mu_g + O(\sigma).
\]
Using smooth approximations of the cutoff function on $\Sigma_\epsilon$, we observe that
\[
\left(\int_{\Sigma_\epsilon} \lambda^\sigma d\mu_g \right)^{\frac 12} \leq \left(\frac{27\pi}4\right)^{\frac 12} \int_{\Gamma_\epsilon}(\lambda^\sigma)^{\frac 12}d\sigma_{\Gamma_\epsilon} + O(\sigma),
\]and letting $\sigma \to 0$ we see that
\[
\left(\int_{\Sigma_\epsilon} \lambda d\mu_g \right)^{\frac 12} \leq \left(\frac{27\pi}4\right)^{\frac 12} \int_{\Gamma_\epsilon}\lambda^{\frac 12}d\sigma_{\Gamma_\epsilon}.
\]By the choice of $\Gamma_\epsilon$,
\[
\mathrm{Area}^{\frac 12}\left[ u\left( \Sigma \right) \right] \leq \left( \frac{27 \pi}{4} \right)^{\frac{1}{2}} \int_{\partial \Sigma} \lambda^{\frac{1}{2}} d\sigma_{\partial \Sigma} + 2 \epsilon,
\]which implies the result.
\end{proof}
\section{Proof of the Main Theorem}\label{BTC}
This section consists of three subsections. In Section \ref{ConvSec}, we prove convergence results that produce the limit map and are applied iteratively to produce the bubble maps. Section \ref{MapsSec} contains a description of the bubble tree and the bubble maps. Finally, in Section \ref{NecSec} we prove the no-neck property and energy quantization result, which finishes the proof of Theorem \ref{MAIN}.
\subsection{Convergence Results}\label{ConvSec}
\begin{lemma}\label{BT1}
Let $u_k:(M,g) \to (X,d)$ be a sequence of harmonic maps such that $E\left[ u_k,M \right]<\Lambda<\infty$ and let $\epsilon_{\mathrm{gap}}$ be as in Proposition \ref{GapThm}. Then there exists a subsequence $\{u_k\}$ and a set of points $\{x_1, \dots, x_\ell\}$ with corresponding masses $\{m_1, \dots, m_\ell\}$ where $\ell \leq \Lambda/\epsilon_{\mathrm{gap}}$, and a harmonic map $u:M \to X$ such that
\begin{enumerate}
\item \label{bct1}$u_k \to u$ in $C^0$ uniformly on compact sets in $M \backslash \{x_1, \dots, x_\ell\}$.
\item \label{bct2}For any open subset $\Omega$ with $\overline{\Omega} \subset M \backslash \{x_1, \dots, x_\ell\}$,
\[
\lim_{k \to \infty}E\left[ u_k, \Omega \right]=E\left[ u,\Omega \right].
\]
\item \label{bct3} For all $r>0$ and all $i \in \{1, \dots, \ell\}$,
\[
\lim_{r \to 0} \; \lim_{k \to \infty}E\left[ u_k, D_r(x_i) \right]:=m_i \geq \epsilon_{\mathrm{gap}}.
\]
\item \label{bct4} The energies satisfy the relation
\[
\lim_{k \to \infty} E[u_k] = E[u] + \sum_{i=1}^\ell m_i.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
Items \eqref{bct1} and \eqref{bct3} follow by standard arguments using Proposition \ref{BTtools}.
For \eqref{bct2}, choose $r'>0$ such that for all $x \in M$, $u(D_{r'}(x)) \subset \mathcal B_{\tau(X)/2}(u(x))$. Let $d_\Omega:= d_g(\partial \Omega, \{x_1,\dots, x_\ell\})$. There exists $K_{\Omega} \in \mathbb N$ such that for all $k \geq K_{\Omega}$ and, $x \in \Omega$, $0<r<d_\Omega/2$, $u_k(D_r(x)) \subset \mathcal B_{3\tau(X)/4}(u(x))$. Suppose to the contrary that the energy drops in the limit. Then there exists $y \in \Omega$ and $0<t<d_\Omega/2$ such that $\liminf_{k\to \infty}E\left[ u_k, D_t(x) \right]> E\left[ u,D_t(x) \right]$. But this contradicts the proof of compactness of minimizers in \cite[Lemma 2.3]{Banff1} and thus the claim holds.
Finally, \eqref{bct4} follows immediately from \eqref{bct2} and standard arguments.
\end{proof}
Fix a constant
\begin{equation}\label{eq:CR}
0<C_R \leq \min\left\{\frac \pi 3, \frac{\epsilon_{\mathrm{gap}}}2, C_{\mathrm{mon}}\frac{ \tau^2(X)}{16}\right\}.
\end{equation}
Here $C_{\mathrm{mon}}$ is the monotonicity constant given in \cite[Theorem 3.4]{Banff2}.
To understand the importance of the constants chosen in the following lemma, we provide a brief outline of their significance going forward. Let $r= \min\{d_g(x_i, x_j),i \neq j\}$, for $x_i$ from Lemma \ref{BT1}. In each ball $B_r(x_i)$, there are three regions of interest. In fact these regions will be on the pull back of this domain to $T_{x_i}M$ by the exponential map. We refer to this domain as $B_r(0) \subset T_xM$.
In the next lemma, we choose these regions by choosing $\epsilon_{k,i}, \lambda_{k,i}, c_{k,i}$ where
\[
B_{k \lambda_{k,i}}(c_{k,i}) \subset B_{\epsilon_{k,i}}(c_{k,i}) \subset B_r(0).
\]By construction $\epsilon_{k,i}/(k\lambda_{k,i}) \to \infty$ so the annular scale is not uniform. The scales and how we choose them will help us to complete our main theorem. On the outer region $B_r(0) \backslash B_{\epsilon_{k,i}}(c_{k,i})$, $u_k \to u$ uniformly in $C^0$, where $u$ is a harmonic map. On the inner region $B_{k \lambda_{k,i}}(c_{k,i})$, after an appropriate conformal transformation of the domain, the new maps $\overline u_{k,i}$ will converge to a ``bubble map" - a harmonic map from $\mathbb S^2$. The intermediate region $B_{\epsilon_{k,i}}(c_{k,i})\backslash B_{k \lambda_{k,i}}(c_{k,i})$ is called the ``neck region". The behavior of the sequence $u_k$ on this region is not captured by $u$ or by any of the bubble maps. Thus, the main objective is to determine whether any of the limiting information about $u_k$ escapes in the neck regions. Our main theorem demonstrates that no energy or image is lost in these necks.
\begin{lemma}\label{BT2}Consider a single bubble point $x_i$ with mass $m_i$. For simplicity we denote them in what follows as $x, m$.
There exists a further subsequence, and constants $\epsilon_k \searrow 0$ and $C>0$ such that, identifying $u_k$ with $\exp_x^*u_k$ and $D_k:= D_{2\epsilon_k}(0) \subset T_xM$,
\begin{enumerate}
\item \label{BT21}$E\left[ u_k, D_k \backslash D_{\epsilon_k/8k^2}(0) \right] \to 0$.
\item $u_k(\partial D_k) \subset \mathcal B_{C/k}(u(x))$.
\item for $c_k:=(c_k^1,c_k^2)$ where
\[
c_k^j= \frac {\int_{D_k} x^j |\nabla u_k|^2dx}{\int_{D_k}|\nabla u_k|^2 dx}, \quad \quad |c_k| \leq \epsilon_k/2k^2.
\]
\item \label{BT24} for
\[
\lambda_k := \min\{\lambda: \int_{D_{\epsilon_k}(0)\backslash D_{\lambda}(c_k)} |\nabla u_k|^2 dx \leq C_R\},\quad \quad \lambda_k \leq \epsilon_k/k^2.
\]
\end{enumerate}
\end{lemma}
Note that the proof of this lemma will not require $C^1$ convergence of $u_k$ to $u$, but instead uses the weaker convergence given by items \eqref{bct2}, \eqref{bct3} in Lemma \ref{BT1}.
\begin{proof}Let $\rho_0:=\frac 12 \min\{\mathrm{dist}(x_j, x): j \in \{1, \dots, \ell\}, x_j \neq x\}$. Choose $\epsilon_k \leq \min\{\frac 1k, \rho_0, \mathrm{inj}(M)\}$ to be the largest number such that
\[
\int_{D_k}|\nabla u|^2 \leq \frac m{16k^2}.
\]
We determine the subsequence inductively. For each $k \geq 1$, let $A_k:= D_k \backslash D_{\epsilon_k/8k^2}(0)$. With $\Omega:= \overline A_k$ fixed, items \eqref{bct2}, \eqref{bct3} of Lemma \ref{BT1} imply that there exists $N_k$ such that for all $n \geq N_k$,
\begin{equation}\label{eq:annular}
\int_{A_k}|\nabla u_n|^2 \leq 2\int_{A_k}|\nabla u|^2< \frac m{8k^2}
\end{equation}and
\begin{equation}\label{eq:disk}
\frac{8k^2-1}{8k^2}m\leq \int_{D_{\epsilon_k/8k^2}(0)}|\nabla u_n|^2 \leq \frac{8k^2+1}{8k^2}m.
\end{equation}Moreover, we may increase $N_k$ if necessary so that for all $n \geq N_k$,
\begin{equation}\label{eq:boundarydist}
\sup_{y \in \partial D_k} d(u_n(y), u(y)) \leq \frac 1k.
\end{equation}Set $n_k = \max\{N_k , 1+n_{k-1}\}$. Then the first item follows from \eqref{eq:annular}. The existence of $C$ such that the second item holds follows from the Lipschitz regularity of $u$ combined with \eqref{eq:boundarydist}. The estimates on $c_k, \lambda_k$ follow from \eqref{eq:annular}, \eqref{eq:disk} and their definitions (cf. \cite[Section 6]{Parker}).
\end{proof}
We will need a conformal transformation of $D_k$ onto $S_k \subset \mathbb S^2$ such that $c_k \mapsto (0,0,1)$ and $\partial D_{\lambda_k}(c_k)$ maps to the equator. Let $\pi_{S^2}: \mathbb S^2 \left(\subset \mathbb R^3\right) \to \mathbb R^2 \cong T_x M$ be a fixed stereographic projection that maps the equator to the unit circle and the north pole to the origin. Let $\Psi_k: \mathbb R^2 \to \mathbb R^2$ be given by
\[
\Psi_k (x) := \lambda_k x + c_k.
\]
Then, the map $\Theta_k := \left( \Psi_k \circ \pi_{S^2} \right)^{-1} = \pi_{S^2}^{-1} \circ \Psi_k^{-1}$ is a conformal transformation under which $c_k \mapsto (0,0,1)$ and $\partial D_{ \lambda_k}(c_k)$ maps to the equator. Define $S_k := \Theta_k \left( D_{2\epsilon_k} (0) \right)$. Now let
\begin{equation}\label{eq:bar-u}
\overline u_k:S_k \to X
\end{equation}
be defined as $\overline u_k \circ \Theta_k (x) = u_k (x)$ for all $x \in D_k$. Applying Proposition \ref{BTtools} to the maps $\overline u_k$ we obtain a result analogous to Lemma \ref{BT1} for maps from domains exhausting $\mathbb S^2$. For ease of notation, let $D_r^{\mathbb S^2}(y)$ denote a geodesic disk in $\mathbb S^2$ of radius $r$ and centered at $y \in \mathbb S^2$.
\begin{lemma}\label{BT3}Let $S^-_k$ represent the portion of $S_k$ in the southern hemisphere, $p^-$ denote the south pole, and $\epsilon_{\mathrm{gap}}$ be as in Proposition \ref{GapThm}.
There exists a further subsequence $\{\overline u_k\}$ of harmonic maps with $\overline u_k:S_k \to X$, a harmonic map $\overline u:\mathbb S^2 \to X$, a collection of points $\{y_1, \dots, y_l\}\subset \mathbb S^2$ with corresponding masses $\{m_1, \dots, m_l\}$ such that
\begin{enumerate}
\item \label{BT31} $\overline u_k \to \overline u$ in $C^0$ uniformly on compact subsets of $\mathbb S^2 \backslash \{y_1, \dots, y_l, p^-\}$.
\item \label{BT33} $\lim_{k \to \infty}E\left[ \overline u_k,S_k \right] = m$.
\item \label{BT32} $\lim_{k \to \infty}E\left[ \overline u_k, S_k^- \right]=C_R$.
\item \label{BT34} for any open set $\Omega$ with $\overline{\Omega} \subset \mathbb S^2 \backslash \{y_1, \dots, y_l, p^-\}$,
\[\lim_{k \to \infty}E\left[ \overline u_k, \Omega \right] = E\left[ \overline u, \Omega \right].
\]
\item \label{BT35} for all $r>0$ and $j \in \{1, \dots, l\}$,
\[
\lim_{r \to 0} \; \lim_{k \to \infty} E\left[ \overline u_k, D_r^{\mathbb S^2}(y_j) \right] :=m_j \geq \epsilon_{\mathrm{gap}}.
\]
\item \label{BT35b} there exists $\tau(p^-) \geq 0$ such that
\[
\lim_{k \to \infty} E[\overline u_k,S_k] = E[\overline u,\mathbb S^2] + \tau(p^-)+ \sum_{i=1}^l m_i
\]
\item \label{BT35b2} the map $|\nabla \overline u|^2 + \tau(p^-)\delta_{p^-} + \sum_{i=1}^l m_i\delta_{y_i}$ has center of mass on the $z$-axis.
\item \label{BT35c} if $E[\overline u,\mathbb S^2]<\epsilon_{\mathrm{gap}}$ then $E[\overline u,\mathbb S^2]=0$. In this case, $l>0$ and if $l=1$ then $\tau(p^-)=C_R$.
\item \label{BT36} $\overline u_k(\partial \Theta_k(D_{k\lambda_k}(c_k))) \subset \mathcal B_{C/k}(\overline u(p^-))$.
\item \label{BT37}
$ E \left[ \overline{u}_k , \Theta_k \left( D_{2k\lambda_k}(c_k) \backslash D_{k\lambda_k}(c_k) \right) \right] \to 0$.
\end{enumerate}
\end{lemma}
\begin{remark}
Following the usual convention, if $E[\overline u,\mathbb S^2]=0$ then we say that $\overline u$ is a \emph{ghost bubble}.
\end{remark}
\begin{proof}Item \eqref{BT31} follows from arguments as in Lemma \ref{BT1} and item \eqref{BT33} follows from the choice of $D_k$ earlier. Observe also that
\begin{align*}
E\left[ \overline u_k, S_k^- \right] = E\left[ u_k, D_k \backslash D_{\lambda_k}(c_k) \right]
=E\left[ u_k, D_k \backslash D_{\epsilon_k}(0) \right] + E\left[ u_k, D_{\epsilon_k}(0)\backslash D_{\lambda_k}(c_k) \right].
\end{align*} Item \eqref{BT32} now follows from Lemma \ref{BT1} \eqref{bct2} and Lemma \ref{BT2} \eqref{BT21}, \eqref{BT24}. Items \eqref{BT34} -- \eqref{BT35b} follow the logic as in Lemma \ref{BT2}, though we must include $\tau(p^-)$ in \eqref{BT35b} since energy may concentrate at $p^-$.
Item \eqref{BT35b2} holds since for $f \in C^\infty(\mathbb S^2, \mathbb R)$, by approximating by characteristic functions and appealing to the logic that gives \eqref{BT35b}, we conclude that
\[
\lim_{k \to \infty}\int_{\mathbb S^2 \cap S_k} f|\nabla \overline u_k|^2 d\mu_g = \int_{\mathbb S^2} f|\nabla \overline u|^2 d\mu_g + f(p^-) \tau(p^-)+ \sum_{i=1}^l f(y_i)m_i.
\]
The first part of item \eqref{BT35c} is immediate from the gap theorem. In that case, $l>0$ by items \eqref{BT33} and \eqref{BT32} and the fact that the $y_j$'s are in the northern hemisphere. When $l=1$, items \eqref{BT32} and \eqref{BT35b2} and the fact that $y_1$ must be in the northern hemisphere imply the result on $\tau(p^-)$. Item \eqref{BT36} follows as in Lemma \ref{BT2}. For item \eqref{BT37}, first notice that
\[
E \left[ \overline{u} , \Theta_k \left( D_{2k\lambda_k}(c_k) \right) \right] \to 0 \quad as \quad k \to \infty.
\]
By item \eqref{BT34}, for each fixed $k\geq 1$ we can choose $N_k$ such that for all $n \ge N_k$,
\[
\left| E \left[ \overline{u}_n , \Theta_k \left( D_{2k\lambda_k} (c_k) \backslash D_{k\lambda_k}(c_k) \right) \right] - E \left[ \overline{u} , \Theta_k \left( D_{2k\lambda_k}(c_k) \backslash D_{k\lambda_k} (c_k)\right) \right] \right|< \frac{1}{k}.
\]
Letting $n_k := \max\{n_{k-1}+1 , N_k\}$, we see that
\[
\lim_{k \to \infty}E \left[ \overline{u}_{n_k} , \Theta_k \left( D_{2k\lambda_k} (c_k) \backslash D_{k\lambda_k} (c_k)\right) \right] \to 0.
\]
Renaming the sequence implies item \eqref{BT37}.
\end{proof}
\subsection{The bubble tree}\label{MapsSec}
Given a bubble point $x_i \in M$ from Lemma \ref{BT1}, by Lemma \ref{BT3} the maps $\overline u_{k,i}:S_{k,i} \to X$ converge to a map $\overline u_i:\mathbb S^2 \to X$ except at $\{y_{i1}, \dots, y_{il_i}, p^-\}$. The process then iterates at each $y_{ij}$ which allows us to obtain bubbles on bubbles. By Lemma \ref{BT3}, item \eqref{BT35c}, there can be at most $\min\{\Lambda/C_R, \log_2(\Lambda/\epsilon_{\mathrm{gap}})\}$ ghost bubbles. Since every non-ghost bubble contains at least $\epsilon_{\mathrm{gap}}$ energy, the process terminates.
Prior to constructing the bubble tree, we prove two technical facts. First, given image curves $\gamma (\partial D_r)$ with small length and energy bounded by $C\delta/r$, there exists a coning off of $\gamma$ in $X$ which has energy bounded by $C\delta$. Second, the sequence of maps $u_k$ possess such curves on scale $\epsilon_k$ and $k\lambda_k$. These technical facts will be useful in the construction of our bubble tree.
\subsubsection{Coning off a curve}
We first demonstrate the existence of a coning off of a Lipschitz curve with energy depending on the energy of the curve.
\begin{definition}\label{def:coneoff}
Let $\gamma: \partial D_r \to \mathcal B_{\tau(X)}(P) \subset X$ be a Lipschitz map. We define the \emph{cone extension map} $\mathrm{Cone}(\gamma_{\partial D_r}): D_r \to X$ such that
\[
\mathrm{Cone}(\gamma_{\partial D_r})(s,\theta) = \eta_\theta \left( \frac{s}{r} \right),
\] where $\eta_\theta:[0,1] \to X$ is the constant speed geodesic connecting $c_\gamma$ to $\gamma(\theta)$ and $c_\gamma$ is the circumcenter of $\gamma$.
\end{definition}
\begin{lemma}\label{lem:coneoff}
Let $\left( D_{2r} , ds^2 + \mu^2(s,\theta)s^2 d\theta^2 \right)$ be a smooth disk such that $s^{-2}|1-\mu^2| + s^{-1} |\partial \mu^2| + |\partial^2 \mu^2| \leq c< \frac 14$ and $r<1$. Let $X$ be a compact locally CAT(1) space with injectivity radius $\tau(X)$. There exist $C, \delta >0$ depending on $X$ such that the following holds: for any Lipschitz loop $\gamma: \partial D_r \to X$ with $E[\gamma, \partial D_r] < \frac{\delta}{r} $, the cone extension map $\mathrm{Cone}(\gamma_{\partial D_r}): D_r \to X$ exists and satisfies $E[\mathrm{Cone}(\gamma_{\partial D_r}),D_r] \le C\delta$.
\end{lemma}
\begin{proof}By the Cauchy-Schwarz inequality,
\[
\mathrm{Length}^2(\gamma) \leq 2\pi \int_0^{2\pi} \left|\frac{\partial \gamma}{\partial \theta}\right|^2 d\theta \leq 2\pi \mu(r, \theta) \delta< \frac{\tau^2(X)}2
\]for sufficiently small $\delta$. Thus $\gamma \subset \mathcal B_{C\delta^{1/2}}(c_\gamma)$ and $\mathrm{Cone}(\gamma_{\partial D_r})(D_r) \subset \mathcal B_{C\delta^{1/2}}(c_\gamma)$. For convenience, let $u:=\mathrm{Cone}(\gamma_{\partial D_r})$.
By the $L-$convexity of CAT(1) spaces \cite[Definition 2.6 and Proposition 3.1]{ohta}, we deduce that
\[
d\left( \eta_{\theta_1}(t) , \eta_{\theta_2}(t) \right) \le \left( 1 + C \delta \right) t\, d \left( \gamma \left( \theta_1 \right) , \gamma\left( \theta_2 \right) \right).
\]
Now we estimate the directional derivative $\left| u_*\left( {\partial_\theta} \right) \right|\left( s,\theta_0 \right) $. Let $\mu := \mu (s , \theta_0)$ and let $\| \cdot \|$ denote the distance to $0$ with respect to the metric $g = ds^2 + \mu^2s^2d\theta^2$. Then by \cite[Section 1.9]{korevaar-schoen1}, for a.e. $(s, \theta_0) \in D_r$,
\begin{align*}
\left| u_*\left( {\partial}_{ \theta} \right) \right|\left( s,\theta_0 \right) &= \lim_{h \to 0}\frac{d\left( u\left( s,\theta_0 \right) , u\left( \mathrm{exp}_{(s,\theta_0)}\left(h\partial_\theta \right) \right) \right) }{|h|} \notag \\
&\le \lim_{h \to 0} \frac{d\left( u\left( s,\theta_0 \right) , u\left( \|\mathrm{exp}_{(s,\theta_0)}\left({h\partial_\theta} \right) \| , \theta_0 \right) \right)}{|h|} \notag \\
&+ \lim_{h \to 0} \frac{d\left( u \left(\|\mathrm{exp}_{(s,\theta_0)}\left(h\partial_\theta\right) \|,\theta_0 \right) , u \left(\|\mathrm{exp}_{(s,\theta_0)}\left(h{\partial_\theta} \right) \|, \mathrm{arg}_\theta\left( \mathrm{exp}_{(s,\theta_0)}\left(h{\partial_\theta} \right) \right) \right) \right) }{|h|}. \notag
\end{align*}
As the radial geodesics are constant speed, and using the $L-$convexity estimate, we deduce that
\begin{align*}
\left| u_*\left({\partial_\theta} \right) \right| \left( s,\theta_0 \right) & \le \frac{d\left(c_\gamma , \gamma\left( \theta_0 \right) \right) }r \lim_{h \to 0} \frac{\| \mathrm{exp}_{(s,\theta_0)}\left(h{\partial_\theta}\right) \| - s}{|h|} \notag \\ &+ \lim_{h \to 0}\left( 1 + C \delta \right) \frac{ \| \mathrm{exp}_{(s,\theta_0)}\left(h{\partial_\theta}\right) \| }{r}\; \frac{d \left( \gamma\left( \theta_0 \right) , \gamma \left( \mathrm{arg}_\theta\left( \mathrm{exp}_{(s,\theta_0)}\left(h{\partial_\theta}\right) \right) \right) \right) }{ |h|}. \notag
\end{align*}
Since ${\partial_s}$ and ${\partial_\theta}$ are perpendicular, the first variation formula implies that
\[
\lim_{h \to 0} \frac{\| \mathrm{exp}_{(s,\theta_0)}\left(h{\partial_\theta} \right) \| - s}{ |h|} = 0.
\]
Thus,
\[
\| \mathrm{exp}_{(s,\theta_0)}\left(h{\partial_\theta}\right) \| = s + o(|h|)|h|.
\]Let $\Delta \theta(h):= \mathrm{arg}_\theta\left( \mathrm{exp}_{(s,\theta_0)}\left(h{\partial_\theta}\right) \right)-\theta_0$. Then,
\begin{align*}
\lim_{h \to 0} \frac{d \left( \gamma\left( \theta_0 \right) , \gamma \left( \theta_0+ \Delta \theta(h) \right) \right) }{h} &= \lim_{h \to 0} \frac{d \left( \gamma\left( \theta_0 \right) , \gamma \left( \theta_0+ \Delta \theta(h) \right) \right) }{\Delta \theta(h)} \cdot \lim_{h \to 0} \frac{\Delta \theta(h)}{h}\\
&= \left| \frac{d \gamma}{d\theta} \right|_{\theta = \theta_0} \cdot \mu.
\end{align*}
Therefore, the directional derivative $\left| u_*\left( \frac{1}{\mu s} \frac{\partial}{\partial \theta} \right) \right|$ satisfies
\begin{equation*}
\frac{1}{\mu s}\left|u_*(\partial_\theta)\right| = \left| u_*\left( \frac{1}{\mu s}{\partial_\theta} \right) \right|\left( s,\theta_0 \right) \le \frac{1 + C \delta}r\left| \frac{d \gamma}{d\theta} \right|_{\theta = \theta_0} .
\end{equation*}
Moreover, one easily calculates, using the constant speed of $\eta$,
\[
\left| u_*(\partial_s) \right| \left( s , \theta_0 \right) = \frac{1}{r} d\left( c_\gamma , \gamma\left(\theta_0 \right) \right) \le \frac{C\delta^{1/2}}{r}.
\]
It follows that
\begin{align*}
\left| \nabla u \right|^2\left( s , \theta_0\right)
&=\left| u_*({\partial_s} )\right|^2(s,\theta_0)+ \frac {1}{\mu^2 s^2} \left| u_*\left( {\partial_\theta} \right) \right|^2(s,\theta_0)\\
&\leq \frac {C^2\delta}{r^2} + \frac{( 1 + C \delta)^2}{r^2}\left| \frac{d \gamma}{d\theta}\right|^2(\theta_0).
\end{align*}
Increasing $C$ as necessary,
\begin{align*}
E[u,D_r] &= \int_{D_r} \left| \nabla u \right|^2\left( s , \theta\right) \mu(s,\theta) s ds d\theta\\
& \leq \frac{C^2\delta}{r^2}\int_{D_r} \mu(s,\theta)\, s\,ds d\theta + \frac{\left( 1 + C\delta \right)^2}{r^2}\int_{D_r} \left| \frac{d \gamma}{d\theta} \right|^2 \mu(s,\theta)\, s\, ds\, d\theta \\
&\le 2\pi C^2 \delta(1+cr^2) + (1+C\delta)^2 (1+cr^2)^2r E[\gamma, \partial D_r]\\
& \leq C \delta.
\end{align*}
\end{proof}
\subsubsection{Curves with small length}
In subsection Section \ref{NecSec}, we prove the no-neck property and energy quantization using the isoperimetric inequality which applies to conformal harmonic maps. Thus, we want to find scales with small image length for a conformal suspension of $u_k$ which in turn implies small image length for the original maps $u_k$.
We begin by recalling a modification of the previous conformalization scheme (see \cite[Theorem 2.3.4]{jost}).
\begin{lemma}\label{lem:conformal-suspension-2}
Let $u: D_1 \to X$ be a harmonic map with $E\left[ u , D_1 \right] \le \Lambda$. Then, there exists a conformal harmonic map $\tilde{u}: D_1 \to X \times \mathbb{C}$ of $u$ and a universal constant $c_1>0$ such that for all $x \in D_{1/2}$,
\begin{equation*}
\left| \nabla \tilde{u} (x)\right|^2 \le \left| \nabla u (x)\right|^2+ 1 + c_1^2 \Lambda^2.
\end{equation*}
\end{lemma}
\begin{proof}
We construct $\nu: D_1 \to \mathbb C$ to satisfy
\begin{equation*}
\begin{cases}
\partial_{{z}} \overline \nu = 1 & \quad in \quad D_1,\\
\partial_z \nu = -\frac{1}{4}\Phi_u & \quad in \quad D_1, \\
\Delta \nu = 0 & \quad in \quad D_1.
\end{cases}
\end{equation*}
To do this, let $\Psi$ be a holomorphic function with $\partial_z \Psi = -\frac{1}{4} \Phi_u$ where $\Phi_u$ is the Hopf function. Since $\Phi_u$ is holomorphic, $\nu(z) := \overline{z} + \Psi(z) $ satisfies the above conditions. Moreover, $\Phi_\nu = 4 \partial_z \nu \partial_z \overline{\nu} = - \Phi_u$. Let $\tilde u:D_1 \to X \times \mathbb C$ such that $\tilde u(x):= (u(x), \nu(x))$. By construction,
\begin{equation*}
\Phi_{\tilde{u}} = \Phi_u + \Phi_\nu = \Phi_u - \Phi_u = 0
\end{equation*}and thus $\tilde u$ is conformal.
Since $\Phi_u \in L^1\left( D_1 \right)$ is holomorphic on $D_1$, using the Cauchy integral formula there exists $c_1>0$ such that for all $x \in D_{1/2}$,
\begin{equation*}
\left| \Phi_u(x) \right| \le 4c_1 \Lambda.
\end{equation*}
Therefore, for all $x \in D_{1/2}$,
\begin{equation*}
\left| \nabla \tilde{u} \right|^2 = \left| \nabla u \right|^2 + \left| \nabla \nu \right|^2 = \left| \nabla u \right|^2 + 1 + \frac{\left| \Phi_u \right|^2}{16} \le \left| \nabla u \right|^2 + 1 + {c_1^2}\Lambda^2.
\end{equation*}
\end{proof}
\begin{definition}
Henceforth we refer to the above constructed $\tilde u$ as the \emph{conformal suspension} of $u$.
\end{definition}
\begin{lemma}\label{lem:susp-area}
Let $u$ and $\tilde{u}$ be as in Lemma~\ref{lem:conformal-suspension-2}. For any $ \Omega \subset D_{1/2}$ and any $0 < r <\frac 12$,
\begin{equation*}
2 \mathrm{Area} \left[ \tilde{u} \left(\Omega \right) \right] = E\left[ \tilde{u} , \Omega \right] \le E\left[ u ,\Omega \right] + \mathrm{Area}\left[\Omega\right] \left( 1 + {c_1^2}\Lambda^2 \right)
\end{equation*}
and
\begin{equation*}
\mathrm{length} \left[ \tilde{u} (\partial D_{r}) \right] \le \mathrm{length} \left[u (\partial D_{r}) \right] + \mathrm{length}[\partial D_r] \left( 1 + {c_1}\Lambda \right).
\end{equation*}Note that length and area of domain regions are taken with respect to the metric $g$.
\end{lemma}
\begin{proof}
Since $\tilde{u}$ is conformal, twice its area coincides with the total energy therefore,
\begin{align*}
2 \mathrm{Area} \left[ \tilde{u} \left(\Omega \right] \right) &= E\left[ \tilde{u} , \Omega \right] = \int_{\Omega} \; \left| \nabla \tilde{u} \right|^2 d\mu_g\\ &\le \int_{\Omega} \;\left( \left| \nabla u \right|^2+ 1 + {c_1^2}\Lambda^2 \right) d\mu_g\\ &= E\left[ u ,\Omega \right] + \mathrm{Area}\left[\Omega \right] \left( 1 + {c_1^2}\Lambda^2 \right).
\end{align*}
Similarly, letting $d\sigma_r$ denote the length measure on $\partial D_r$,
\begin{align*}
\mathrm{length} \left[ \tilde{u} (\partial D_{r}) \right] &= \int_{\partial D_{r}} \; \left| \partial_\theta \tilde{u} \right| \; d\sigma_r = \int_{\partial D_{r}} \; \left| \partial_\theta u \right| + \left| \partial_\theta \nu \right| \; d\sigma_r \\ &\le \mathrm{length} \left[ u (\partial D_{r}) \right] + \int_{\partial D_r} \; 1 + {c_1} \Lambda \; d\sigma_r \\ &= \mathrm{length} \left[ u (\partial D_{r}) \right] + \mathrm{length}[\partial D_r]\left( 1 + {c_1}\Lambda \right).
\end{align*}
\end{proof}
Let $x$ be a fixed bubble point and choose $\rho_0<\frac 12$ so that $D_{2 \rho_0}(x)$ does not contain any other bubble points. Let $\tilde u_k$ denote the conformal suspension of each $u_k|_{D_{2\rho_0}(x)}$ as in Lemma \ref{lem:conformal-suspension-2}. Recall that $\epsilon_k, \lambda_k$ are chosen in Lemma \ref{BT2} and an outline of their significance is given in the paragraph preceding that lemma. The next lemma provides precise scales, comparable to $\epsilon_k$ and $k \lambda_k$, on which we can apply our cone extension lemma. These two scales will determine the boundary of the neck region.
\begin{lemma}\label{lem:length}There exist sequences $r_k \in [\epsilon_k/4, \epsilon_k/2]$ and $s_k \in [k\lambda_k, 2 k \lambda_k]$ such that
\[
\lim_{k \to \infty}{r_k}E[\tilde u_k, \partial D_{r_k}(c_k)] = 0,
\]
\[
\lim_{k \to \infty}s_k E[\tilde u_k, \partial D_{s_k}(c_k)] = 0.
\]As a consequence,
\[
\lim_{k \to \infty}\mathrm{length}[\tilde u_k(\partial D_{r_k}(c_k))] =0,
\]
\[
\lim_{k \to \infty}\mathrm{length}[\tilde u_k(\partial D_{s_k}(c_k))] =0.
\]
\end{lemma}
\begin{proof}
As $\epsilon_k \to 0$, for each map $\tilde u_k$ we consider the metric in the tangent space $(D_{2\epsilon_k}(0), ds^2 + \mu^2_k(s,\theta)s^2 d\theta^2)$ where $s^{-2}|1- \mu_k^2|+s^{-1} |\partial \mu_k^2| + |\partial^2 \mu_k^2| \leq \alpha_k$ where $\alpha_k \to 0$. Let $r_k \in [\epsilon_k/4, \epsilon_k/2]$ such that $E[\tilde u_k, \partial D_{r_k}(c_k)] = \min_{r \in [\epsilon_k/4, \epsilon_k/2]}E[\tilde u_k, \partial D_{r}(c_k)] $. Then
\begin{align*}
\frac {\epsilon_k}4E[\tilde u_k, \partial D_{r_k}(c_k)] &\leq \int_{\epsilon_k/4}^{\epsilon_k/2}\int_0^{2\pi} \frac 1{s \mu_k}\left|\frac{\partial \tilde u_k}{\partial \theta}\right|^2 d\theta ds \\
&\leq E\left[ \tilde u_k, D_{\epsilon_k/2}(c_k) \backslash D_{\epsilon_k/4}(c_k) \right]\\
& \leq E\left[ u_k, D_k \backslash D_{\epsilon_k/8k^2}(0) \right] + \mathrm{Area}\left[ D_k \right] \left(1+ {c_1^2}\Lambda^2 \right)
\end{align*}where the last inequality follows from Lemma \ref{lem:susp-area} and the fact that $ D_{\epsilon_k/2}(c_k) \backslash D_{\epsilon_k/4}(c_k)\subset D_k \backslash D_{\epsilon_k/8k^2}(0)$. By Item \eqref{BT21} of Lemma \ref{BT2} and the fact that $\mathrm{Area}\left[ D_k \right] \leq c \epsilon_k^2$, the final expression tends to zero in $k$. Since $r_k/2 \leq {\epsilon_k}/4$, the desired result follows.
To find the $s_k$'s we use item \eqref{BT37} of Lemma~\ref{BT3} in place of item \eqref{BT21} of Lemma \ref{BT2} and follow a similar reasoning as above.
The length estimates then follow immediately from Cauchy-Schwarz.
\end{proof}
\subsubsection{The base, neck, and bubble maps}
Around each bubble point $x_i \in \{x_1, \dots, x_\ell\}$, there are three domains of interest. In $D_k(x_i) = D_{2\epsilon_{k,i}}(0)\subset T_{x_i}M$, we consider the disks $D_{r_{k,i}}(c_{k,i}), D_{s_{k,i}}(c_{k,i})$ and the annulus between them
\[
A_{k,i}':=D_{r_{k,i}}(c_{k,i})\backslash D_{s_{k,i}}(c_{k,i}).
\]
Here $\epsilon_{k,i}, \lambda_{k,i}, c_{k,i}$ are given by Lemma \ref{BT2} and $r_{k,i}, s_{k,i}$ are given by Lemma \ref{lem:length}.
We define the \emph{neck maps} $u_{k,i}|_{ A_{k,i}'}:A_{k,i}' \to X$. To define the \emph{extended base maps}, let
\[
\underline u_k(x):= \left\{ \begin{array}{ll} u_k(x) & \text{if } x \in M \backslash \cup_{i=1}^\ell \mathrm{exp}(D_{r_{k,i}}(c_{k,i}))\\
\mathrm{Cone}(u_k|_{\partial D_{r_{k,i}}(c_{k,i})}) & \text{if } x \in \mathrm{exp}(D_{r_{k,i}}(c_{k,i})).
\end{array}\right.
\]By Lemmas \ref{BT1}, \ref{lem:length}, and \ref{lem:coneoff}, $\underline u_k \to u$ in $C^0$ uniformly on $M$ and
\[
\lim_{k \to \infty} E[\underline u_k, M] = E[u, M].
\]
Similarly, the \emph{extended bubble maps} will cone off the maps $\overline u_{k,i}$. Let $\underline{\overline u}_{k,i}:\mathbb S^2 \to X$ such that
\[
\underline{\overline u}_{k,i}(x):= \left\{ \begin{array}{ll}
\overline u_{k,i}(x) & \text{if } x \in \Theta_{k,i} (D_{s_{k,i}}(c_{k,i}))\\
\mathrm{Cone}(\overline u_{k,i}|_{\Theta_{k,i} ( \partial D_{s_{k,i}}(c_{k,i}))}(x) &\text{otherwise}.
\end{array}\right.
\]
By Lemmas \ref{BT3}, \ref{lem:length}, and \ref{lem:coneoff}, for each $i \in \{1 , \dots, \ell\}$, $\underline{\overline u}_{k,i} \to {\overline u}_{i}$ uniformly in $C^0$ on $\mathbb S^2 \backslash \{y_{i1}, \dots y_{il_i}\}$ and
\[
\lim_{k \to \infty} E[\underline{\overline u}_{k,i} ,\mathbb S^2] = E[ {\overline u}_i,\mathbb S^2] + \sum_{j=1}^{l_i} m_{ij}.
\]
Note that the term $\tau_i(p^-)$ is lacking from the above limit and the uniform convergence happens across $p^-$. This occurs since we have removed the neck map portion from the extended bubble maps and replaced it by the coning off which has energy and diameter converging to zero. Moreover, the energy contained in the neck maps is exactly
\[
\tau_i(p^-) = \lim_{k \to \infty} E[u_{k,i}, A_{k,i}'].
\] We will show in the next subsection that $\tau_i(p^-)=0$ and $\mathrm{diam}(u_{k,i}(A_{k,i}')) \to 0$ and thus the $C^0$ limit and the limit of the energies of the extended bubble maps are the same as the limit for the original maps.
As mentioned previously, the extended bubble map process now iterates and we construct maps $\underline {\overline u}_{k,ij}:\mathbb S^2 \to X$ where $j \in \{1, \dots, l_i\}$. These maps converge, away from finitely many points $y_{ijm}$, $m \in \{1, \dots, l_{ij}\}$, to some map $ {\overline u}_{ij}:\mathbb S^2 \to X$ with an analogous energy limit to what we saw above.
\subsubsection{Constructing the bubble tree}\label{BTconst}
We now construct the bubble tree and the bubble domain. The bubble tree consists of vertices and edges where each vertex represents a harmonic map and each edge represents a bubble point. The base vertex of the tree is the map $u:M \to X$ and the $\ell$ edges emanating from the base vertex are the points $x_i$. The edges $x_i$ connect to the vertices $ {\overline u}_{i}:\mathbb S^2 \to X$ and the edges emanating from each of these vertices are the bubble points $y_{ij}$, $j \in \{1, \dots, l_i\}$. The tree is finite as the process terminates.
The bubble tower is the disjoint union of $M$ and a collection of $\mathbb S^2$'s, where each $\mathbb S^2$ is associated with a vertex in the bubble tree. Indeed, following \cite{Parker}, we may consider a \emph{bubble tower} $T$ in the following manner. Let $SM$ be an $\mathbb S^2$ bundle over $M$. Compactifying the vertical tangent space of $S M \to M$ yields an $\mathbb S^2$ bundle $S^2M$ over $SM$. Iterating this process then yields a tower of $\mathbb S^2$ fibrations. For clarity, the point $y_{i_1i_2\dots i_n}$ lies in $S^{n-1}M$.
A \emph{bubble domain} at level $n$ is a fiber $F$ of $ S^nM \to S^{n-1}M$ and a \emph{bubble tower} is a finite union of bubble domains such that the projection of $T \cap S^nM$ lies in $T \cap S^{n-1}M$.
Given the sequence $u_k:M \to X$, applying Lemmas \ref{BT1}, \ref{BT2}, \ref{BT3} determines a unique bubble tower $T= M \bigcup \left(\cup_I \mathbb S^2_I\right)$ where $I$ is indexed over all of the bubble points in the process. We define a sequence of bubble tower maps $\overline{\underline u}_{k,I}:T \to X$ such that ${\underline u}_k:M \to X$ and $\overline{\underline u}_{k,I}:\mathbb S^2_I \to X$. Letting $u, \overline {\underline u}_I$ index the limit maps, observe that
\begin{equation}
\lim_{k \to \infty}E[\overline{\underline u}_{k,I},T] = E[\overline{\underline u}_I,T]
\end{equation}and $\overline{\underline u}_{k,I} \to \overline{\underline u}_I$ in $C^0$ uniformly on $T$.
\subsection{Energy quantization and the no neck property}\label{NecSec}
In this subsection, we study the neck maps and use the isoperimetric inequality to prove that the energy of neck maps vanish in the limit. Then by monotonicity, the diameter of the neck maps must also vanish. Taken with the previous subsections, these results immediately imply Theorem \ref{MAIN}.
Consider a single neck map $u_k: A_k' \to X$ where $A_k':=D_{r_k}(c_k) \backslash D_{s_k}(c_k)$.
\begin{lemma}[Vanishing Neck Energy and Length] The following holds:
\[
\limsup_{k \to \infty} E\left[ u_k, A_k' \right] =0,
\]
\[
\limsup_{k \to \infty}\mathrm{diam}\left[u_k(A_k')\right] =0.
\]
\end{lemma}
\begin{proof} Let $\tilde u_k$ denote the conformal suspension of each $u_k|_{D_{r}(x)}$ as in Lemma \ref{lem:conformal-suspension-2}.
By \eqref{eq:CR} and Lemma \ref{lem:susp-area}, for all sufficiently large $k$, $\mathrm{Area}\left[\tilde{u}_k(A_k') \right]< \frac \pi 3$.
By Lemma \ref{lem:length}, for any $0<\delta\leq \tau(X)/4$ there exists a $K$ such that for all $k \geq K$ there exist points $P_k, Q_k \in X \times \mathbb C$ such that $\tilde u_k(\partial A_k') \subset \mathcal B_\delta(P_k) \cup \mathcal B_\delta(Q_k)$. Now suppose that there exists $R_k \in \tilde u_k(A_k')$ such that $R_k \notin
\mathcal B_{2\delta}(P_k) \cup \mathcal B_{2\delta}(Q_k)$. Then, applying the monotonicity formula of \cite[Theorem 3.4]{Banff2} to $\tilde u_k(A_k') \cap \mathcal B_\delta(R_k)$,
\[
C_{\mathrm{mon}} \delta^2 \leq \mathrm{Area}\left[\tilde u_k(A_k')\right] .
\]On the other hand, by Lemma \ref{lem:susp-area}, using the fact that $A_k' \subset D_{\epsilon_k}(0)\backslash D_{\lambda_k}(c_k)$, and recalling the definition of $C_R$ from \eqref{eq:CR},
\[
2\mathrm{Area}\left[\tilde u_k(A_k')\right] \leq E[u_k, D_{\epsilon_k}(0)\backslash D_{\lambda_k}(c_k)]+ \mathrm{Area}[A_k'](1+c_1^2\Lambda^2)\leq C_{\mathrm{mon}}\frac{\tau^2(X)}{16}+ \frac {C}{k^2}(1+c_1^2\Lambda^2).\]
This implies a contradiction for $\delta = \tau(X)/4$ and $k$ sufficiently large. It follows that for $k$ large enough, $\tilde u_k( A_k') \subset \mathcal B_{\tau(X)}(P_k)$. Thus each $\tilde u_k:A_k' \to X$ satisfies the hypotheses of the isoperimetric inequality, Theorem \ref{thm:isoperimetric}. By Lemma \ref{lem:length}, it thus follows that
\[
E\left[ u_k, A_k' \right] \leq E\left[ \tilde u_k,A_k' \right] = 2\mathrm{Area}\left[\tilde u_k(A_k')\right] \leq \frac{27\pi}2 \mathrm{length}^2\left[\tilde u_k(\partial A_k')\right]\to 0.
\]
With this improvement on the area estimate, for any fixed $\delta>0$ we may choose $N$ large enough so that for all $k \geq N$,
$\mathrm{Area}\left[\tilde u_k(A_k')\right] < C_{\mathrm{mon}}\delta^2/2$ and there exist points $P_k, Q_k \in X \times \mathbb C$ such that $\tilde u_k(\partial A_k') \subset \mathcal B_\delta(P_k) \cup \mathcal B_\delta(Q_k)$. If there exists $R_k \in \tilde u_k(A_k')$ such that $R_k \notin
\mathcal B_{2\delta}(P_k) \cup \mathcal B_{2\delta}(Q_k)$ then by the same argument as above, the monotonicity formula implies a contradiction. Therefore, $\tilde u_k(A_k') \subset \mathcal B_{4\delta}(P_k)$. It follows that
\[
\lim_{k \to \infty} \mathrm{diam}\left[u_k(A_k')\right] \leq \lim_{k \to \infty} \mathrm{diam}\left[\tilde u_k(A_k')\right]=0.
\]
\end{proof}
|
{
"timestamp": "2018-02-27T02:07:46",
"yymm": "1802",
"arxiv_id": "1802.08905",
"language": "en",
"url": "https://arxiv.org/abs/1802.08905"
}
|
"\\section{I. Introduction}\n\n\n\n\n\n\n\nThe conformal bootstrap is the idea that a conformally in(...TRUNCATED)
| {"timestamp":"2019-01-03T02:00:37","yymm":"1802","arxiv_id":"1802.08911","language":"en","url":"http(...TRUNCATED)
|
"\\section{Mathematical setting}\nIn this section we introduce the basic tools.\n\n\\subsection{The (...TRUNCATED)
| {"timestamp":"2018-12-14T02:10:42","yymm":"1802","arxiv_id":"1802.08623","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\nRecent small-scale experiments \\cite{barends2014, corcoles2015, kelly20(...TRUNCATED)
| {"timestamp":"2018-02-26T02:12:40","yymm":"1802","arxiv_id":"1802.08680","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction \\label{sec:Intro}}\n\nThe analysis of big data is one of the most important(...TRUNCATED)
| {"timestamp":"2018-09-25T02:26:43","yymm":"1802","arxiv_id":"1802.08703","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\\label{sec:intro}\n\nGalaxies, due to complexities inherent to their format(...TRUNCATED)
| {"timestamp":"2018-12-07T02:00:59","yymm":"1802","arxiv_id":"1802.08694","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\nTopological semimetals (TSMs) were predicted\ntheoretically~\\cite{Burko(...TRUNCATED)
| {"timestamp":"2018-02-26T02:11:56","yymm":"1802","arxiv_id":"1802.08643","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}}\n\\label{sec:mp_intro}\n\n\\IEEEPARstart{O}{wing} to the advent of online (...TRUNCATED)
| {"timestamp":"2018-02-27T02:06:40","yymm":"1802","arxiv_id":"1802.08869","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\n\\label{intro}\n\n\\textit{Mesic nuclei} are currently one of the hottes(...TRUNCATED)
| {"timestamp":"2018-03-30T02:05:42","yymm":"1802","arxiv_id":"1802.08597","language":"en","url":"http(...TRUNCATED)
|
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