2601/2601.00648.md

Title: Lipschitz Stability for an Inverse Problem of Biharmonic Wave Equations with Damping

URL Source: https://arxiv.org/html/2601.00648

Markdown Content: 1Introduction 2Properties of the operator 𝒜 3Well-posedness and Observability 4Proof of the Main Theorems Lipschitz Stability for an Inverse Problem of Biharmonic Wave Equations with Damping Minghui Bi School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, P.R.China. bimh801@nenu.edu.cn Yixian Gao School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024, P.R.China. gaoyx643@nenu.edu.cn Abstract.

This paper establishes Lipschitz stability for the simultaneous recovery of a variable density coefficient and the initial displacement in a damped biharmonic wave equation. The data consist of the boundary Cauchy data for the Laplacian of the solution, Δ ​ 𝑢 | ∂ Ω and ∂ 𝑛 ( Δ ​ 𝑢 ) | ∂ Ω . We first prove that the associated system operator generates a contraction semigroup, which ensures the well-posedness of the forward problem. A key observability inequality is then derived via multiplier techniques. Building on this foundation, explicit stability estimates for the inverse problem are obtained. These estimates demonstrate that the biharmonic structure inherently enhances the stability of parameter identification, with the stability constants exhibiting an explicit dependence on the damping coefficient via the factor ( 1 + 𝛾 ) 1 / 2 . This work provides a rigorous theoretical basis for applications in non-destructive testing and dynamic inversion.

Key words and phrases: Biharmonic Wave Equations, Parameter Identification, Inverse Problem, Stability 2010 Mathematics Subject Classification: 37K55, 35Q30, 76D05 The research of Y. Gao was supported by NSFC grant 12371187 and STDP Project of Jilin Province 20240101006JJ 1.Introduction

The biharmonic wave equation constitutes a fundamental model for the vibration of thin elastic plates. A principal inverse problem arising in this context involves the simultaneous determination of both internal material parameters, such as a spatially heterogeneous density, and the initial state of the system from boundary measurements.

We consider this inverse problem for a damped biharmonic wave equation with variable density 𝜌 ​ ( 𝑥 ) on a bounded domain Ω ⊂ ℝ 𝑛

{ 𝜌 ​ ( 𝑥 ) ​ ∂ 𝑡 2 𝑢 ​ ( 𝑡 , 𝑥 ) + Δ 2 ​ 𝑢 ​ ( 𝑡 , 𝑥 ) + 𝛾 ​ ∂ 𝑡 𝑢 ​ ( 𝑡 , 𝑥 )

0 ,

( 𝑡 , 𝑥 ) ∈ ℝ + × Ω ,

𝑢 ​ ( 0 , 𝑥 )

𝑓 ​ ( 𝑥 ) , ∂ 𝑡 𝑢 ​ ( 0 , 𝑥 )

𝑔 ​ ( 𝑥 ) ,

𝑥 ∈ Ω ,

𝑢 ​ ( 𝑡 , 𝑥 )

0 , ∂ 𝑢 ∂ 𝑛 ​ ( 𝑡 , 𝑥 )

0 ,

( 𝑡 , 𝑥 ) ∈ ℝ + × ∂ Ω .

(1.1)

The model assumes a piecewise constant density function 𝜌 ​ ( 𝑥 ) ; a typical example takes the form (as in [16]):

𝜌 ​ ( 𝑥 )

𝜌 0 + ( 𝜌 1 − 𝜌 0 ) ​ 𝟏 𝜔 ​ ( 𝑥 ) ,

where 𝜔 ⊂ Ω is a star-shaped inclusion and 𝜌 0 , 𝜌 1

0 . Here, the constant 𝛾 ≥ 0 denotes the damping coefficient, and Δ 2 represents the biharmonic operator. The functions 𝑓 and 𝑔 correspond to the initial displacement and velocity, respectively.

We consider the corresponding inverse problem formulated in terms of boundary measurements of the trace

Δ ​ 𝑢 | ∂ Ω , ( ∂ Δ ​ 𝑢 ) ∂ 𝑛 | ∂ Ω ,

collected over a finite time interval [ 0 , 𝑇 ] with 𝑇

0 . Our main objective is to establish Lipschitz-type stability estimates for the simultaneous recovery of the unknown density coefficient 𝜌 and the initial displacement 𝑓 from this boundary data.

To formulate the problem precisely, we introduce the following class of admissible parameters. The variable density 𝜌 is assumed to satisfy

0 < 𝜌 min ≤ 𝜌 ​ ( 𝑥 ) ≤ 𝜌 max < ∞ , 𝑥 ∈ Ω .

The initial displacement 𝑓 belongs to the space 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) and satisfies the compatibility conditions

𝑓 | ∂ Ω

0 , ∂ 𝑓 ∂ 𝑛 | ∂ Ω

0 , Δ 2 ​ 𝑓 | ∂ Ω

0 ,

which guarantee 𝐻 4 -regularity of the solution. The initial velocity 𝑔 is taken from 𝐻 0 2 ​ ( Ω ) . A triple ( 𝜌 , 𝑓 , 𝑔 ) satisfying these conditions is called admissible.

Our main stability result can now be stated as follows.

Theorem 1.1.

Let 𝑢 1 and 𝑢 2 be the solutions of system (1.1) corresponding to two admissible parameter sets ( 𝜌 1 , 𝑓 1 , 𝑔 1 ) and ( 𝜌 2 , 𝑓 2 , 𝑔 2 ) , and denote by 𝑢 := 𝑢 1 − 𝑢 2 . Assume that the acceleration of the second solution satisfies

‖ ∂ 𝑡 2 𝑢 2 ‖ 𝐿 ∞ ​ ( 0 , 𝑇 ; 𝐿 2 ​ ( Ω ) ) ≤ 𝑀

for some constant 𝑀 > 0 . Then there exists a constant 𝐶

𝐶 ​ ( 𝑀 , 𝑇 , 𝜌 min , 𝜌 max , Ω )

0 such that

‖ 𝜌 1 − 𝜌 2 ‖ 𝐿 ∞ ​ ( Ω ) ≤ 𝐶 ​ ( 1 + 𝛾 ) 1 / 2 ​ ( ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ d 𝑆 ​ d 𝑡 ) 1 / 2 .

(1.2)

Theorem 1.1 shows that the 𝐿 2 -norm of the boundary observation controls the difference of the densities in the 𝐿 ∞ -norm.

Theorem 1.2.

Under the assumptions of Theorem 1.1, and assuming further that 𝑔 1

𝑔 2 , suppose the observation time 𝑇 satisfies the geometric condition

𝑇

2 ​ diam ​ ( Ω ) 𝜌 min .

Then there exists a constant 𝐶

𝐶 ​ ( 𝑀 , 𝑇 , 𝜌 min , 𝜌 max , Ω )

0 such that

‖ 𝑓 1 − 𝑓 2 ‖ 𝐻 4 ​ ( Ω ) ≤ 𝐶 ​ ( 1 + 𝛾 ) 1 / 2 ​ ( ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ d 𝑆 ​ d 𝑡 ) 1 / 2 ,

where 𝑢

𝑢 1 − 𝑢 2 .

This result demonstrates that, when the initial velocities coincide, the boundary measurements also control the difference of the initial displacements in the 𝐻 4 -norm. The stability is a consequence of the energy observability inequality, the well-posedness of the forward problem, and the regularity properties of the biharmonic operator.

The mathematical study of wave equations originated in the early twentieth century. Foundational contributions included the construction of special solutions via auxiliary equations [20, 19, 6, 14, 12], the analysis of solution convergence for problems with singular boundaries, the effect of damping terms on well-posedness, and the establishment of uniqueness theorems. Together, these works laid the essential groundwork for the subsequent development of higher-order wave models.

Research on the biharmonic wave equation emerged from classical boundary value problems. In the context of the half-plane biharmonic equation with elliptic holes, Mackevič [3] introduced an elliptic coordinate system along with a family of functions 𝐶 , establishing the existence and uniqueness of solutions under prescribed regularity conditions on the hole boundaries, zero Dirichlet conditions on the straight boundaries, and certain directional derivative constraints. This foundational result paved the way for later work on both forward and inverse problems. Subsequent developments extended the analysis to more general settings: Atakhodzhaev [3] constructed approximate solutions for three-dimensional biharmonic equations via regularization; Arzhanykh [2] proved the existence of finitely many quasi-analytic solutions in three dimensions; and Ramm [21] resolved the half-plane Cauchy problem using Fourier transform methods.

Recent progress in the study of biharmonic inverse problems has followed a trajectory from fundamental theoretical advances to nonlinear generalizations and progressively broader application settings. In [13, 18], the authors utilized complex geometric optics (CGO) solutions and Fourier analysis to derive sharp stability estimates for the density inversion in damped fourth-order plate equations, achieving the simultaneous reconstruction of both the density and an internal source via time-frequency transformations. In the nonlinear regime, Acosta [1] applied high-frequency transformation theory to related inversion problems. Subsequently, Feizmohammadi [11] relaxed structural assumptions for the nonhomogeneous wave equation, thereby enabling the unique recovery of multiple coefficients from boundary data. This line of inquiry was extended by Kian [15] to one-dimensional evolution equations, where coefficients are determined solely from boundary measurements. Further broadening the scope, Chitorkin [9] generalized multi-dimensional wave equation inversion to unbounded domains and established local solvability and stability for inverse Sturm–Liouville problems with singular potentials using single-measurement data.

The core difficulties of biharmonic inverse problems [22, 17, 5, 10, 7, 4] correspond directly to several significant and persistent gaps in the literature. The higher-order nature of the operator significantly complicates well-posedness analysis, while spatially dependent density and damping terms further increase the challenges of stable reconstruction. Additionally, rigorous stability estimates for the simultaneous recovery of multiple parameters remain scarce, and quantifying the relationship between limited boundary observations and unknown internal parameters presents a formidable analytical challenge. Consequently, most existing results are confined to single-parameter inversion or simplified governing equations, leaving a clear gap in the simultaneous reconstruction of both density and initial displacement, as well as in the derivation of explicit stability estimates for the variable-density, damped fourth-order biharmonic wave equation.

To address these challenges, the present work establishes several fundamental advances. We move beyond the conventional setting of single-parameter recovery by providing a framework for the simultaneous reconstruction of the medium’s density and the initial displacement. The well-posedness of the forward problem is established rigorously by proving that the associated system operator generates a contraction semigroup. Using multiplier methods, we derive a novel energy observability inequality, which forms the analytical cornerstone for the subsequent inverse analysis. Our main contribution is the first explicit Lipschitz stability estimate for this coupled inverse problem, precisely quantifying how limited boundary observations control the unknown parameters. These results provide a rigorous theoretical foundation for applications in non-destructive evaluation and confirm the intrinsic stability afforded by the biharmonic structure. Collectively, this work advances the theory of higher-order inverse problems and introduces a coherent analytical framework for stable multi-parameter reconstruction in complex wave models.

The paper is organized as follows. In Section 2, we establish the well-posedness of the forward problem by analyzing the system operator 𝒜 in the natural energy space ℋ . Section 3 derives a key observability inequality via the multiplier method, which forms the analytical foundation for the inverse stability analysis. Finally, Section 4 contains the proofs of Theorems 1.1 and 1.2, where Lipschitz stability estimates for the density and initial displacement are obtained by comparing two solutions and applying the observability inequality systematically.

2.Properties of the operator 𝒜

To establish the well-posedness of the forward problem for system (1.1), we introduce the appropriate energy space and system operator. Define the energy space

ℋ

( 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) ) × 𝐻 0 2 ​ ( Ω ) ,

equipped with the inner product

⟨ ( 𝑢 1 , 𝑣 1 ) , ( 𝑢 2 , 𝑣 2 ) ⟩ ℋ

∫ Ω Δ 2 ​ 𝑢 1 ⋅ Δ 2 ​ 𝑢 2 ¯ ​ d 𝑥 + ∫ Ω 𝜌 ​ 𝑣 1 ​ 𝑣 2 ¯ ​ d 𝑥 ,

which corresponds to the total energy of the system. The second term involves the weighted space 𝐿 2 ​ ( Ω ; 𝜌 ​ d ​ 𝑥 ) with norm

‖ 𝑢 ‖ 𝐿 2 ​ ( Ω ; 𝜌 ​ d ​ 𝑥 ) 2

∫ Ω 𝜌 ​ ( 𝑥 ) ​ | 𝑢 ​ ( 𝑥 ) | 2 ​ d 𝑥 .

The system operator 𝒜 : 𝐷 ​ ( 𝒜 ) ⊂ ℋ → ℋ is defined by

𝒜 ​ ( 𝑢

𝑣 )

( 𝑣

− 𝜌 − 1 ​ Δ 2 ​ 𝑢 − 𝜌 − 1 ​ 𝛾 ​ 𝑣 ) ,

with domain

𝐷 ​ ( 𝒜 )

{ ( 𝑢 , 𝑣 ) ∈ ( 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) ) × 𝐻 0 2 ​ ( Ω ) | − 𝜌 − 1 ​ Δ 2 ​ 𝑢 − 𝜌 − 1 ​ 𝛾 ​ 𝑣 ∈ 𝐿 2 ​ ( Ω ) } .

The well-posedness of the corresponding abstract Cauchy problem follows from fundamental properties of 𝒜 . We now prove that 𝒜 is maximally dissipative with dense domain, which ensures the existence of a contraction semigroup via the Hille–Yosida theorem.

Lemma 2.1 (Dissipativity).

The operator 𝒜 is dissipative, that is, for every ( 𝑢 , 𝑣 ) ∈ 𝐷 ​ ( 𝐴 ) ,

Re ⟨ 𝒜 ( 𝑢 , 𝑣 ) , ( 𝑢 , 𝑣 ) ⟩ ℋ ≤ 0 .

Proof.

For ( 𝑢 , 𝑣 ) ∈ 𝐷 ​ ( 𝒜 ) , we computer

⟨ 𝒜 ​ ( 𝑢 , 𝑣 ) , ( 𝑢 , 𝑣 ) ⟩ ℋ

⟨ ( 𝑣

− 𝜌 − 1 ​ Δ 2 ​ 𝑢 − 𝜌 − 1 ​ 𝛾 ​ 𝑣 ) , ( 𝑢

𝑣 ) ⟩ ℋ

∫ Ω Δ 2 ​ 𝑣 ​ Δ 2 ​ 𝑢 ¯ ​ d 𝑥 + ∫ Ω 𝜌 ​ ( − 𝜌 − 1 ​ Δ 2 ​ 𝑢 − 𝜌 − 1 ​ 𝛾 ​ 𝑣 ) ​ 𝑣 ¯ ​ d 𝑥

∫ Ω Δ 2 ​ 𝑣 ​ Δ 2 ​ 𝑢 ¯ ​ d 𝑥 − ∫ Ω Δ 2 ​ 𝑢 ​ 𝑣 ¯ ​ d 𝑥 − 𝛾 ​ ∫ Ω | 𝑣 | 2 ​ d 𝑥 .

Let

𝐼

∫ Ω Δ 2 ​ 𝑣 ​ Δ 2 ​ 𝑢 ¯ ​ d 𝑥 − ∫ Ω Δ 2 ​ 𝑢 ​ 𝑣 ¯ ​ d 𝑥 .

Taking the complex conjugate yields

𝐼 ¯

∫ Ω Δ 2 ​ 𝑣 ¯ ​ Δ 2 ​ 𝑢 ​ d 𝑥 − ∫ Ω Δ 2 ​ 𝑢 ¯ ​ 𝑣 ​ d 𝑥

− 𝐼 ,

which shows that Re ⁡ ( 𝐼 )

0 . Consequently,

Re ⟨ 𝒜 ( 𝑢 , 𝑣 ) , ( 𝑢 , 𝑣 ) ⟩ ℋ

− 𝛾 ∫ Ω | 𝑣 | 2 d 𝑥 ≤ 0 ,

establishing the dissipativity of 𝒜 . ∎

Dissipativity alone is insufficient to guarantee that an operator generates a 𝐶 0 -semigroup. According to the Hille–Yosida theorem, a necessary additional condition is that the operator be maximally dissipative; i.e., that ran ⁡ ( 𝜆 ​ 𝐼 − 𝒜 )

ℋ for some 𝜆

0 . We now verify that our operator 𝒜 satisfies this property.

Lemma 2.2 (Maximality).

The system operator 𝒜 : 𝐷 ​ ( 𝒜 ) ⊂ ℋ → ℋ is maximally dissipative.

Proof.

Lemma 2.1 establishes the dissipativity of 𝒜 . To verify maximal dissipativity, it suffices by the Hille–Yosida theorem to show that ran ⁡ ( 𝜆 ​ 𝐼 − 𝒜 )

ℋ for some 𝜆

0 . We show in fact that this holds for every 𝜆

0 .

Fix 𝜆 > 0 and let 𝐲

( 𝑦 1 , 𝑦 2 ) ∈ ℋ . We seek 𝐱

( 𝑢 , 𝑣 ) ∈ 𝐷 ​ ( 𝒜 ) satisfying

( 𝜆 ​ 𝐼 − 𝒜 ) ​ 𝐱

𝐲 .

Writing out 𝒜 explicitly yields the system

𝜆 ​ ( 𝑢

𝑣 ) − ( 𝑣

− 𝜌 − 1 ​ Δ 2 ​ 𝑢 − 𝜌 − 1 ​ 𝛾 ​ 𝑣 )

( 𝑦 1

𝑦 2 ) ,

which is equivalent to

𝜆 ​ 𝑢 − 𝑣

𝑦 1 ,

(2.1a)

𝜆 ​ 𝑣 + 𝜌 − 1 ​ Δ 2 ​ 𝑢 + 𝜌 − 1 ​ 𝛾 ​ 𝑣

𝑦 2 .

(2.1b)

From (2.1a) we have 𝑣

𝜆 ​ 𝑢 − 𝑦 1 . Substituting this into (2.1b) gives

𝜆 ​ ( 𝜆 ​ 𝑢 − 𝑦 1 ) + 𝜌 − 1 ​ Δ 2 ​ 𝑢 + 𝜌 − 1 ​ 𝛾 ​ ( 𝜆 ​ 𝑢 − 𝑦 1 )

𝑦 2 .

Rearranging terms produces an elliptic equation for 𝑢

𝜌 − 1 ​ Δ 2 ​ 𝑢 + ( 𝜆 2 + 𝜌 − 1 ​ 𝛾 ​ 𝜆 ) ​ 𝑢

𝑦 2 + 𝜆 ​ 𝑦 1 + 𝜌 − 1 ​ 𝛾 ​ 𝑦 1 .

Multiplying by 𝜌 ​ ( 𝑥 ) (recalling that 𝜌 ​ ( 𝑥 ) ≥ 𝜌 min

0 ) yields the final elliptic problem

Δ 2 ​ 𝑢 + 𝜇 ​ 𝜌 ​ 𝑢

𝐹 in  ​ Ω ,

(2.2)

where

𝜇 := 𝜆 2 + 𝜌 − 1 ​ 𝛾 ​ 𝜆

0 , 𝐹 := 𝜌 ​ 𝑦 2 + 𝜌 ​ 𝜆 ​ 𝑦 1 + 𝛾 ​ 𝑦 1 .

Since the coefficient 𝜇 belongs to 𝐿 ∞ ​ ( Ω ) and is strictly positive, and 𝐹 ∈ 𝐿 2 ​ ( Ω ) . Under the clamped boundary conditions 𝑢

∂ 𝑢 / ∂ 𝑛

0 on ∂ Ω , standard elliptic theory for the biharmonic operator guarantees the existence of a unique solution to (2.2)

𝑢 ∈ 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) .

Define 𝑣 := 𝜆 ​ 𝑢 − 𝑦 1 . Since 𝑢 ∈ 𝐻 0 2 ​ ( Ω ) and 𝑦 1 ∈ 𝐻 0 2 ​ ( Ω ) (as 𝐲 ∈ ℋ ), it follows that 𝑣 ∈ 𝐻 0 2 ​ ( Ω ) . Moreover, from (2.2), we have Δ 2 ​ 𝑢

𝑓 − 𝜇 ​ 𝜌 ​ 𝑢 ∈ 𝐿 2 ​ ( Ω ) , and consequently

𝒜 ​ 𝐱

( 𝑣 , − 𝜌 − 1 ​ Δ 2 ​ 𝑢 − 𝜌 − 1 ​ 𝛾 ​ 𝑣 ) ∈ ℋ .

Hence 𝐱

( 𝑢 , 𝑣 ) belongs to 𝐷 ​ ( 𝒜 ) and satisfies ( 𝜆 ​ 𝐼 − 𝒜 ) ​ 𝐱

𝐲 .

Since 𝜆 > 0 and 𝐲 ∈ ℋ were arbitrary, we conclude that ran ⁡ ( 𝜆 ​ 𝐼 − 𝒜 )

ℋ for every 𝜆

0 . Therefore, 𝒜 is maximally dissipative.

∎

We now verify that 𝒜 is closed—another necessary condition for semigroup generation.

Lemma 2.3 (Closedness).

The operator 𝒜 is closed.

Proof.

Let { 𝐱 𝑛 }

{ ( 𝑢 𝑛 , 𝑣 𝑛 ) } ⊂ 𝐷 ​ ( 𝒜 ) be a sequence such that,

𝐱 𝑛 → 𝐱 and 𝒜 ​ 𝐱 𝑛 → 𝐲 in  ​ ℋ ,

as 𝑛 → ∞ , where 𝐱

( 𝑢 , 𝑣 ) and 𝐲

( 𝑦 1 , 𝑦 2 ) .

We show that 𝐱 ∈ 𝐷 ​ ( 𝒜 ) and 𝒜 ​ 𝐱

𝐲 . By the definition of the energy norm, convergence in ℋ means

‖ Δ 2 ​ ( 𝑢 𝑛 − 𝑢 ) ‖ 𝐿 2 ​ ( Ω ) → 0 and ‖ 𝜌 ​ ( 𝑣 𝑛 − 𝑣 ) ‖ 𝐿 2 ​ ( Ω ) → 0 ,

as 𝑛 → ∞ . since 𝜌 ≥ 𝜌 min

0 , the second limit implies

‖ 𝑣 𝑛 − 𝑣 ‖ 𝐿 2 ​ ( Ω ) → 0 .

(2.3)

The convergence of 𝒜 ​ 𝐱 𝑛 in ℋ means

𝒜 ​ 𝐱 𝑛

( 𝑣 𝑛

− 𝜌 − 1 ​ Δ 2 ​ 𝑢 𝑛 − 𝜌 − 1 ​ 𝛾 ​ 𝑣 𝑛 ) → ( 𝑦 1

𝑦 2 )

𝐲 in  ​ ℋ .

This leads to

𝑣 𝑛

→ 𝑦 1 in  ​ 𝐻 0 2 ​ ( Ω ) ,

(2.4)

− 𝜌 − 1 ​ Δ 2 ​ 𝑢 𝑛 − 𝜌 − 1 ​ 𝛾 ​ 𝑣 𝑛

→ 𝑦 2 in  ​ 𝐿 2 ​ ( Ω ) .

(2.5)

From (2.4) and (2.3) we obtain

𝑣

𝑦 1 ∈ 𝐻 0 2 ​ ( Ω ) .

Rewriting (2.5) as

− 𝜌 − 1 ​ Δ 2 ​ 𝑢 𝑛

𝑦 2 + 𝜌 − 1 ​ 𝛾 ​ 𝑣 𝑛 + 𝜖 𝑛 , 𝜖 𝑛 → 0 ​  in  ​ 𝐿 2 ​ ( Ω ) .

Multiplying by − 𝜌 and using (2.3), we obtain

Δ 2 ​ 𝑢 𝑛 → − 𝜌 ​ 𝑦 2 − 𝛾 ​ 𝑣 in  ​ 𝐿 2 ​ ( Ω ) .

Now, the biharmonic operator

Δ 2 : 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) ⊂ 𝐿 2 ​ ( Ω ) → 𝐿 2 ​ ( Ω )

is closed. Since 𝑢 𝑛 → 𝑢 in 𝐻 0 2 ​ ( Ω ) ⊂ 𝐿 2 ​ ( Ω ) and Δ 2 ​ 𝑢 𝑛 converges in 𝐿 2 ​ ( Ω ) , it follows that

𝑢 ∈ 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) and Δ 2 ​ 𝑢

− 𝜌 ​ 𝑦 2 − 𝛾 ​ 𝑣 .

Therefore 𝐱

( 𝑢 , 𝑣 ) ∈ 𝐷 ​ ( 𝒜 ) . Finally,

𝒜 ​ 𝐱

( 𝑣 , − 𝜌 − 1 ​ Δ 2 ​ 𝑢 − 𝜌 − 1 ​ 𝛾 ​ 𝑣 )

( 𝑦 1 , − 𝜌 − 1 ​ ( − 𝜌 ​ 𝑦 2 − 𝛾 ​ 𝑣 ) − 𝜌 − 1 ​ 𝛾 ​ 𝑣 )

( 𝑦 1 , 𝑦 2 )

𝐲 ,

which completes the proof. ∎

The denseness of the domain 𝐷 ​ ( 𝒜 ) in ℋ provides the necessary approximation property to handle general initial data within the semigroup framework.

Lemma 2.4 (Denseness).

The domain 𝐷 ​ ( 𝒜 ) is dense in ℋ ; i.e., 𝐷 ​ ( 𝒜 ) ¯

ℋ .

Proof.

Let 𝐱

( 𝑢 , 𝑣 ) ∈ ℋ be arbitrary. By definition, this means 𝑢 ∈ 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) and 𝑣 ∈ 𝐻 0 2 ​ ( Ω ) . Since 𝐶 0 ∞ ​ ( Ω ) is dense in both 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) and 𝐻 0 2 ​ ( Ω ) with respect to their respective Sobolev norms, there exist sequences { 𝑢 𝑛 } ⊂ 𝐶 0 ∞ ​ ( Ω ) and { 𝑣 𝑛 } ⊂ 𝐶 0 ∞ ​ ( Ω ) such that

‖ 𝑢 𝑛 − 𝑢 ‖ 𝐻 4 ​ ( Ω ) → 0 and ‖ 𝑣 𝑛 − 𝑣 ‖ 𝐻 2 ​ ( Ω ) → 0 as  ​ 𝑛 → ∞ .

Setting 𝐱 𝑛 := ( 𝑢 𝑛 , 𝑣 𝑛 ) , we have 𝐱 𝑛 ∈ 𝐶 0 ∞ ​ ( Ω ) × 𝐶 0 ∞ ​ ( Ω ) ⊂ 𝐷 ​ ( 𝒜 ) by construction.

To establish convergence in ℋ , consider the norm

‖ 𝐱 𝑛 − 𝐱 ‖ ℋ 2

∫ Ω | Δ 2 ​ ( 𝑢 𝑛 − 𝑢 ) | 2 ​ d 𝑥 + ∫ Ω 𝜌 ​ | 𝑣 𝑛 − 𝑣 | 2 ​ d 𝑥 .

For the first term, the continuity of the operator Δ 2 : 𝐻 4 ​ ( Ω ) → 𝐿 2 ​ ( Ω ) yields that there exists a constant 𝐶

0 such that

∫ Ω | Δ 2 ​ ( 𝑢 𝑛 − 𝑢 ) | 2 ​ d 𝑥 ≤ 𝐶 ​ ‖ 𝑢 𝑛 − 𝑢 ‖ 𝐻 4 ​ ( Ω ) 2 → 0 .

For the second term, using the boundedness of 𝜌 (with 𝜌 ≤ 𝜌 max ) and the fact that 𝐻 2 -convergence implies 𝐿 2 -convergence, we obtain

∫ Ω 𝜌 ​ | 𝑣 𝑛 − 𝑣 | 2 ​ 𝑑 𝑥 ≤ 𝜌 max ​ ‖ 𝑣 𝑛 − 𝑣 ‖ 𝐿 2 ​ ( Ω ) 2 → 0 .

Consequently, ‖ 𝐱 𝑛 − 𝐱 ‖ ℋ → 0 as 𝑛 → ∞ , proving that 𝐷 ​ ( 𝒜 ) is dense in ℋ . ∎

In summary, the operator 𝒜 is closed, densely defined, and maximally dissipative on the energy space ℋ . By the Hille–Yosida theorem, it therefore generates a 𝐶 0 -semigroup of contractions { 𝑒 𝑡 ​ 𝒜 } 𝑡 ≥ 0 on ℋ . This establishes the well-posedness of the associated forward problem, which we analyze in the following section.

3.Well-posedness and Observability

This section establishes the well‑posedness of the forward problem and proves a key observability inequality for the subsequent analysis of inverse problems. We first present a fundamental lemma on the equivalence between the abstract energy norm and standard Sobolev norms, which is essential for the ensuing regularity theory and energy estimates. Building on the semigroup framework developed in Section 2, we then prove the existence, uniqueness, and regularity of solutions to system (1.1), together with an exponential energy estimate. Finally, under a geometric control condition, we derive an observability inequality that bounds the full initial energy by boundary measurements. This inequality forms the foundation for the stability analysis of the associated inverse problems.

Lemma 3.1.

Let Ω ⊂ ℝ 𝑛 be a bounded domain with sufficiently smooth boundary, and let 𝜌 ∈ 𝐿 ∞ ​ ( Ω ) satisfy 0 < 𝜌 min ≤ 𝜌 ​ ( 𝑥 ) ≤ 𝜌 max . Then the energy norm on ℋ is equivalent to the standard product norm on 𝐻 4 ​ ( Ω ) × 𝐿 2 ​ ( Ω ) . More precisely, there exist constants 𝐶 1 , 𝐶 2

0 , depending on Ω , 𝜌 min , and 𝜌 max , such that for any ( 𝑓 , 𝑔 ) ∈ ℋ ,

𝐶 1 ​ ( ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 + ‖ 𝑔 ‖ 𝐿 2 ​ ( Ω ) 2 ) ≤ ‖ ( 𝑓 , 𝑔 ) ‖ ℋ 2 ≤ 𝐶 2 ​ ( ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 + ‖ 𝑔 ‖ 𝐿 2 ​ ( Ω ) 2 ) .

Proof.

We establish the norm equivalence by proving the upper and lower bounds separately.

Upper bound: By definition of the energy norm,

‖ ( 𝑓 , 𝑔 ) ‖ ℋ 2

∫ Ω | Δ 2 ​ 𝑓 | 2 ​ d 𝑥 + ∫ Ω 𝜌 ​ | 𝑔 | 2 ​ d 𝑥 .

For the first term, since Δ 2 is a linear differential operator of order 4 , standard elliptic regularity yields a constant 𝐶 Δ 2

0 such that

∫ Ω | Δ 2 ​ 𝑓 | 2 ​ d 𝑥 ≤ 𝐶 ​ ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 .

For the second term, the upper bound on 𝜌 gives

∫ Ω 𝜌 ​ | 𝑔 | 2 ​ d 𝑥 ≤ 𝜌 max ​ ‖ 𝑔 ‖ 𝐿 2 ​ ( Ω ) 2 .

Combining these estimates, we obtain

‖ ( 𝑓 , 𝑔 ) ‖ ℋ 2 ≤ 𝐶 ​ ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 + 𝜌 max ​ ‖ 𝑔 ‖ 𝐿 2 ​ ( Ω ) 2 ≤ max ⁡ ( 𝐶 , 𝜌 max ) ​ ( ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 + ‖ 𝑔 ‖ 𝐿 2 ​ ( Ω ) 2 ) ,

which establishes the upper bound with 𝐶 2

max ⁡ ( 𝐶 , 𝜌 max ) .

Lower bound: For the second term, the lower bound 𝜌 ≥ 𝜌 min

0 implies

∫ Ω 𝜌 ​ | 𝑔 | 2 ​ d 𝑥 ≥ 𝜌 min ​ ‖ 𝑔 ‖ 𝐿 2 ​ ( Ω ) 2 .

(3.1)

For the first term, we apply Gårding’s inequality to the strongly elliptic operator Δ 2 , which provides constants 𝑐 1

0 and 𝑐 2 ≥ 0 such that for every 𝑓 ∈ 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) ,

∫ Ω | Δ 2 ​ 𝑓 | 2 ​ d 𝑥 ≥ 𝑐 1 ​ ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 − 𝑐 2 ​ ‖ 𝑓 ‖ 𝐿 2 ​ ( Ω ) 2 .

(3.2)

Since 𝑓 ∈ 𝐻 0 2 ​ ( Ω ) , we may apply the Poincaré inequality repeatedly. There exists a constant 𝐶 𝑃

0 , depending only on Ω , such that

‖ 𝑓 ‖ 𝐿 2 ​ ( Ω ) ≤ 𝐶 𝑃 2 ​ ‖ 𝐷 2 ​ 𝑓 ‖ 𝐿 2 ​ ( Ω ) ≤ 𝐶 𝑃 4 ​ ‖ 𝐷 4 ​ 𝑓 ‖ 𝐿 2 ​ ( Ω ) ≤ 𝐶 𝑃 4 ​ ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) .

(3.3)

Substituting estimate (3.3) into (3.2) gives

∫ Ω | Δ 2 ​ 𝑓 | 2 ​ d 𝑥

≥ 𝑐 1 ​ ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 − 𝑐 2 ​ ( 𝐶 𝑃 4 ​ ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) ) 2

= ( 𝑐 1 − 𝑐 2 ​ 𝐶 𝑃 8 ) ​ ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 .

The constant 𝑐 1 − 𝑐 2 ​ 𝐶 𝑃 8 is positive for the biharmonic operator on a bounded domain. Setting 𝑐 3 := 𝑐 1 − 𝑐 2 ​ 𝐶 𝑃 8

0 , we obtain

∫ Ω | Δ 2 ​ 𝑓 | 2 ​ d 𝑥 ≥ 𝑐 3 ​ ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 .

(3.4)

Combining (3.1) and (3.4) yields

‖ ( 𝑓 , 𝑔 ) ‖ ℋ 2 ≥ 𝑐 3 ​ ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 + 𝜌 min ​ ‖ 𝑔 ‖ 𝐿 2 ​ ( Ω ) 2 ≥ min ⁡ ( 𝑐 3 , 𝜌 min ) ​ ( ‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) 2 + ‖ 𝑔 ‖ 𝐿 2 ​ ( Ω ) 2 ) ,

(3.5)

which establishes the lower bound with 𝐶 1

min ⁡ ( 𝑐 3 , 𝜌 min ) . This completes the proof. ∎

Equipped with the norm equivalence established in Lemma 3.1, we now turn to the well-posedness of the original system (1.1). The following theorem establishes the existence, uniqueness, regularity and a key energy estimate for the solution, which will be essential for the subsequent stability analysis of the inverse problems.

Theorem 3.2 (Well-posedness and Energy Estimate).

Let Ω ⊂ ℝ 3 be a bounded domain with sufficiently smooth boundary, 𝜌 ∈ 𝐿 ∞ ​ ( Ω ) satisfy 0 < 𝜌 min ≤ 𝜌 ​ ( 𝑥 ) ≤ 𝜌 max , and 𝛾 ≥ 0 . If the initial data ( 𝑓 , 𝑔 ) ∈ 𝐷 ​ ( 𝐴 ) satisfy the compatibility conditions, then the system

{ 𝜌 ​ ( 𝑥 ) ​ ∂ 𝑡 2 𝑢 + Δ 2 ​ 𝑢 + 𝛾 ​ ∂ 𝑡 𝑢

0 ,

( 𝑡 , 𝑥 ) ∈ ℝ + × Ω ,

𝑢 ​ ( 0 , 𝑥 )

𝑓 ​ ( 𝑥 ) , ∂ 𝑡 𝑢 ​ ( 0 , 𝑥 )

𝑔 ​ ( 𝑥 ) ,

𝑥 ∈ Ω ,

𝑢 ​ ( 𝑡 , 𝑥 )

0 , ∂ 𝑢 ∂ 𝑛 ​ ( 𝑡 , 𝑥 )

0 ,

( 𝑡 , 𝑥 ) ∈ ℝ + × ∂ Ω ,

admits a unique solution

𝑢 ∈ 𝐶 ​ ( [ 0 , ∞ ) ; 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) ) ∩ 𝐶 1 ​ ( [ 0 , ∞ ) ; 𝐻 0 2 ​ ( Ω ) ) ∩ 𝐶 2 ​ ( [ 0 , ∞ ) ; 𝐿 2 ​ ( Ω ) ) .

Moreover, there exist constants 𝐶 , 𝐾

0 , depending on Ω , 𝜌 , and 𝛾 , such that the energy satisfies

‖ ( 𝑢 ​ ( 𝑡 ) , ∂ 𝑡 𝑢 ​ ( 𝑡 ) ) ‖ ℋ ≤ 𝐶 ​ 𝑒 𝐾 ​ 𝑡 ​ ‖ ( 𝑓 , 𝑔 ) ‖ ℋ , ∀ 𝑡 ≥ 0 .

Proof.

We divide the proof into three parts: existence, uniqueness, and the energy estimate.

Existence. From Section 2, the operator 𝒜 is maximally dissipative and densely defined in ℋ . By the Hille–Yosida theorem[8], 𝒜 generates a 𝐶 0 -semigroup of contractions { 𝑒 𝑡 ​ 𝒜 } 𝑡 ≥ 0 on ℋ .

For initial data ( 𝑓 , 𝑔 ) ∈ 𝐷 ​ ( 𝒜 ) , define the vector function

𝑼 ​ ( 𝑡 )

𝑒 𝑡 ​ 𝒜 ​ ( 𝑓 , 𝑔 ) ⊤ .

Then 𝑼 is the unique solution of the abstract Cauchy problem

d d ​ 𝑡 ​ 𝑼 ​ ( 𝑡 )

𝒜 ​ 𝑼 ​ ( 𝑡 ) , 𝑼 ​ ( 0 )

( 𝑓 , 𝑔 ) ,

with regularity

𝑼 ∈ 𝐶 ​ ( [ 0 , ∞ ) ; 𝐷 ​ ( 𝒜 ) ) ∩ 𝐶 1 ​ ( [ 0 , ∞ ) ; ℋ ) .

Identifying the components 𝑼 ​ ( 𝑡 )

( 𝑢 ​ ( 𝑡 ) , ∂ 𝑡 𝑢 ​ ( 𝑡 ) ) , the stated Sobolev regularity for 𝑢 follows from the definition of 𝐷 ​ ( 𝒜 ) and the standard semigroup theory.

Uniqueness. Uniqueness is immediate from semigroup theory. For completeness, we provide an alternative energy argument. Let 𝑢 1 and 𝑢 2 be two solutions with the same initial data, and set 𝑤

𝑢 1 − 𝑢 2 . Define the energy

𝐸 𝑤 ​ ( 𝑡 )

1 2 ​ ∫ Ω ( 𝜌 ​ | ∂ 𝑡 𝑤 | 2 + | Δ ​ 𝑤 | 2 ) ​ d 𝑥 .

Differentiating and using the homogeneous equation yields

d d ​ 𝑡 ​ 𝐸 𝑤 ​ ( 𝑡 )

− 𝛾 ​ ∫ Ω | ∂ 𝑡 𝑤 | 2 ​ d 𝑥 ≤ 0 .

Since 𝐸 𝑤 ​ ( 0 )

0 and 𝐸 𝑤 ​ ( 𝑡 ) ≥ 0 , it follows that 𝐸 𝑤 ​ ( 𝑡 ) ≡ 0 , hence 𝑤 ≡ 0 .

Energy estimate. Let 𝑣

∂ 𝑡 𝑢 and define the energy functional

𝐸 ​ ( 𝑡 )

1 2 ​ ‖ ( 𝑢 ​ ( 𝑡 ) , 𝑣 ​ ( 𝑡 ) ) ‖ ℋ 2

1 2 ​ ( ∫ Ω | Δ 2 ​ 𝑢 | 2 ​ 𝑑 𝑥 + ∫ Ω 𝜌 ​ | 𝑣 | 2 ​ d 𝑥 ) .

Differentiating with respect to time and substituting the PDE yields

d ​ 𝐸 d ​ 𝑡

∫ Ω Δ 2 ​ 𝑢 ⋅ Δ 2 ​ 𝑣 ​ d 𝑥 + ∫ Ω 𝜌 ​ 𝑣 ⋅ ∂ 𝑡 2 𝑢 ​ d ​ 𝑥

= ∫ Ω Δ 2 ​ 𝑢 ⋅ Δ 2 ​ 𝑣 ​ d 𝑥 + ∫ Ω 𝑣 ⋅ ( − Δ 2 ​ 𝑢 − 𝛾 ​ 𝑣 ) ​ d 𝑥

= ∫ Ω Δ 2 ​ 𝑢 ⋅ Δ 2 ​ 𝑣 ​ d 𝑥 − ∫ Ω 𝑣 ⋅ Δ 2 ​ 𝑢 ​ d 𝑥 − 𝛾 ​ ∫ Ω | 𝑣 | 2 ​ d 𝑥 .

Set 𝑁 ​ ( 𝑡 )

2 ​ 𝐸 ​ ( 𝑡 )

‖ ( 𝑢 ​ ( 𝑡 ) , 𝑣 ​ ( 𝑡 ) ) ‖ ℋ 2 . Then

d ​ 𝑁 d ​ 𝑡

2 ​ ∫ Ω Δ 2 ​ 𝑢 ⋅ Δ 2 ​ 𝑣 ​ d 𝑥 − 2 ​ ∫ Ω 𝑣 ⋅ Δ 2 ​ 𝑢 ​ d 𝑥 − 2 ​ 𝛾 ​ ∫ Ω | 𝑣 | 2 ​ d 𝑥 .

By the Cauchy-Schwarz and Young inequalities, together with the boundedness of Δ 2 : 𝐻 4 ​ ( Ω ) → 𝐿 2 ​ ( Ω ) , there exists a constant 𝐾 1

0 such that

| 2 ​ ∫ Ω Δ 2 ​ 𝑢 ⋅ Δ 2 ​ 𝑣 ​ d 𝑥 − 2 ​ ∫ Ω 𝑣 ⋅ Δ 2 ​ 𝑢 ​ d 𝑥 | ≤ 𝐾 1 ​ 𝑁 ​ ( 𝑡 ) .

Consequently,

d ​ 𝑁 d ​ 𝑡 ≤ 𝐾 1 ​ 𝑁 ​ ( 𝑡 ) .

Gronwall’s inequality implies 𝑁 ​ ( 𝑡 ) ≤ 𝑁 ​ ( 0 ) ​ 𝑒 𝐾 1 ​ 𝑡 , i.e.,

‖ ( 𝑢 ​ ( 𝑡 ) , ∂ 𝑡 𝑢 ​ ( 𝑡 ) ) ‖ ℋ ≤ 𝑒 𝐾 1 ​ 𝑡 / 2 ​ ‖ ( 𝑓 , 𝑔 ) ‖ ℋ .

Taking 𝐶

1 and 𝜔

𝐾 1 / 2 completes the proof of the energy estimate (3.2). ∎

The stability estimates for the inverse parameter identification problems (Theorems 1.1 and 1.2) rely fundamentally on establishing a quantitative relationship between the internal initial energy of the system and the corresponding boundary measurements. This connection is precisely quantified by an observability inequality. In what follows, we employ the multiplier method under the Geometric Control Condition (GCC) to prove this pivotal result. The essence of the multiplier approach lies in the strategic selection of a vector field to extract energy dissipation and boundary contributions through integration by parts.

Theorem 3.3 (Observability Inequality).

Let Ω ⊂ ℝ 𝑛 be a bounded domain with smooth boundary that is star-shaped with respect to some point 𝑥 0 ∈ Ω , that is,

( 𝑥 − 𝑥 0 ) ⋅ 𝑛 ​ ( 𝑥 ) ≥ 𝑐 0

0 , ∀ 𝑥 ∈ ∂ Ω ,

(3.6)

where 𝑛 ​ ( 𝑥 ) denotes the unit outer normal vector at 𝑥 ∈ ∂ Ω . Let 𝜌 ∈ 𝐿 ∞ ​ ( Ω ) satisfy 0 < 𝜌 min ≤ 𝜌 ​ ( 𝑥 ) ≤ 𝜌 max , and let 𝛾 ≥ 0 . Suppose 𝑢 is a solution of the system

{ 𝜌 ​ ( 𝑥 ) ​ ∂ 𝑡 2 𝑢 + Δ 2 ​ 𝑢 + 𝛾 ​ ∂ 𝑡 𝑢

0 ,

( 𝑡 , 𝑥 ) ∈ ( 0 , 𝑇 ) × Ω ,

𝑢 ​ ( 0 , 𝑥 )

𝑓 ​ ( 𝑥 ) , ∂ 𝑡 𝑢 ​ ( 0 , 𝑥 )

𝑔 ​ ( 𝑥 ) ,

𝑥 ∈ Ω ,

𝑢 ​ ( 𝑡 , 𝑥 )

0 , ∂ 𝑢 ∂ 𝑛 ​ ( 𝑡 , 𝑥 )

0 ,

( 𝑡 , 𝑥 ) ∈ ( 0 , 𝑇 ) × ∂ Ω ,

(3.7)

with initial data ( 𝑓 , 𝑔 ) ∈ ℋ . If the observation time 𝑇 satisfies

𝑇

2 ⋅ diam ⁡ ( Ω ) 𝜌 min ,

(3.8)

then there exists a constant 𝐶

𝐶 ​ ( Ω , 𝜌 , 𝑇 )

0 such that

𝐸 ​ ( 0 ) ≤ 𝐶 ​ ( 1 + 𝛾 ) ​ ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ d 𝑆 ​ d 𝑡

where the initial energy is given by

𝐸 ​ ( 0 )

1 2 ​ ∫ Ω ( 𝜌 ​ ( 𝑥 ) ​ | 𝑔 ​ ( 𝑥 ) | 2 + | Δ ​ 𝑓 ​ ( 𝑥 ) | 2 ) ​ d 𝑥 .

Proof.

We employ the multiplier method with the vector field 𝑚 ​ ( 𝑥 )

𝑥 − 𝑥 0 , which satisfies the star-shaped condition 𝑚 ⋅ 𝑛 ≥ 𝑐 0

0 on ∂ Ω . Multiplying the governing equation by 𝑚 ⋅ ∇ 𝑢 and integrating over [ 0 , 𝑇 ] × Ω yields

∫ 0 𝑇 ∫ Ω ( 𝜌 ​ ∂ 𝑡 2 𝑢 + Δ 2 ​ 𝑢 + 𝛾 ​ ∂ 𝑡 𝑢 ) ​ ( 𝑚 ⋅ ∇ 𝑢 ) ​ d 𝑥 ​ d 𝑡

0 .

Decompose this integral as 𝐼 1 + 𝐼 2 + 𝐼 3

0 , where

𝐼 1

∫ 0 𝑇 ∫ Ω 𝜌 ​ ∂ 𝑡 2 𝑢 ​ ( 𝑚 ⋅ ∇ 𝑢 ) ​ d ​ 𝑥 ​ d ​ 𝑡 ,

𝐼 2

∫ 0 𝑇 ∫ Ω Δ 2 ​ 𝑢 ​ ( 𝑚 ⋅ ∇ 𝑢 ) ​ d 𝑥 ​ d 𝑡 ,

𝐼 3

∫ 0 𝑇 ∫ Ω 𝛾 ​ ∂ 𝑡 𝑢 ​ ( 𝑚 ⋅ ∇ 𝑢 ) ​ d ​ 𝑥 ​ d ​ 𝑡 .

Estimate of 𝐼 1 : We begin by integrating by parts in the time variable. This yields

𝐼 1

[ ∫ Ω 𝜌 ​ ∂ 𝑡 𝑢 ​ ( 𝑚 ⋅ ∇ 𝑢 ) ​ d ​ 𝑥 ] 0 𝑇 − ∫ 0 𝑇 ∫ Ω 𝜌 ​ ∂ 𝑡 𝑢 ​ ∂ 𝑡 ( 𝑚 ⋅ ∇ 𝑢 ) ​ d ​ 𝑥 ​ d ​ 𝑡 .

Since ∂ 𝑡 ( 𝑚 ⋅ ∇ 𝑢 )

𝑚 ⋅ ∇ ​ ∂ 𝑡 𝑢 , we have

∫ Ω 𝜌 ​ ∂ 𝑡 𝑢 ​ ( 𝑚 ⋅ ∇ ​ ∂ 𝑡 𝑢 ) ​ d ​ 𝑥

1 2 ​ ∫ Ω 𝜌 ​ 𝑚 ⋅ ∇ | ∂ 𝑡 𝑢 | 2 ​ d ​ 𝑥 .

Applying the divergence theorem yields

∫ Ω 𝜌 ​ 𝑚 ⋅ ∇ | ∂ 𝑡 𝑢 | 2 ​ d ​ 𝑥

− ∫ Ω ∇ ⋅ ( 𝜌 ​ 𝑚 ) ​ | ∂ 𝑡 𝑢 | 2 ​ d 𝑥 + ∫ ∂ Ω 𝜌 ​ | ∂ 𝑡 𝑢 | 2 ​ ( 𝑚 ⋅ 𝑛 ) ​ d 𝑆 .

The boundary term vanishes due to ∂ 𝑡 𝑢 | ∂ Ω

0 . Consequently,

𝐼 1

[ ∫ Ω 𝜌 ​ ∂ 𝑡 𝑢 ​ ( 𝑚 ⋅ ∇ 𝑢 ) ​ d ​ 𝑥 ] 0 𝑇 + 1 2 ​ ∫ 0 𝑇 ∫ Ω ∇ ⋅ ( 𝜌 ​ 𝑚 ) ​ | ∂ 𝑡 𝑢 | 2 ​ d 𝑥 ​ d 𝑡 .

(3.9)

Estimate of 𝐼 2 : We apply Green’s formula twice to 𝐼 2

∫ 0 𝑇 ∫ Ω Δ 2 ​ 𝑢 ​ ( 𝑚 ⋅ ∇ 𝑢 ) ​ d 𝑥 ​ d 𝑡 . This gives

𝐼 2

∫ 0 𝑇 ∫ Ω Δ ​ 𝑢 ​ Δ ​ ( 𝑚 ⋅ ∇ 𝑢 ) ​ d 𝑥 ​ d 𝑡 + ∫ 0 𝑇 ∫ ∂ Ω [ ∂ Δ ​ 𝑢 ∂ 𝑛 ​ ( 𝑚 ⋅ ∇ 𝑢 ) − Δ ​ 𝑢 ​ ∂ ∂ 𝑛 ​ ( 𝑚 ⋅ ∇ 𝑢 ) ] ​ d 𝑆 ​ d 𝑡 .

(3.10)

Since ∇ 𝑚

𝐼 and Δ ​ 𝑚

0 , we compute Δ ​ ( 𝑚 ⋅ ∇ 𝑢 )

𝑚 ⋅ ∇ ( Δ ​ 𝑢 ) + 2 ​ Δ ​ 𝑢 . Substituting this into the volume integral and performing a further integration by parts leads to

𝐼 2

( 2 − 𝑛 2 ) ​ ∫ 0 𝑇 ∫ Ω | Δ ​ 𝑢 | 2 ​ d 𝑥 ​ d 𝑡 + 1 2 ​ ∫ 0 𝑇 ∫ ∂ Ω | Δ ​ 𝑢 | 2 ​ ( 𝑚 ⋅ 𝑛 ) ​ d 𝑆 ​ d 𝑡

• ∫ 0 𝑇 ∫ ∂ Ω [ ∂ Δ ​ 𝑢 ∂ 𝑛 ​ ( 𝑚 ⋅ ∇ 𝑢 ) − Δ ​ 𝑢 ​ ∂ ∂ 𝑛 ​ ( 𝑚 ⋅ ∇ 𝑢 ) ] ​ d 𝑆 ​ d 𝑡 .

(3.11)

On the boundary ∂ Ω , the clamped conditions 𝑢

0 and ∂ 𝑢 / ∂ 𝑛

0 imply 𝑚 ⋅ ∇ 𝑢

0 . Moreover, a direct calculation shows

∂ ∂ 𝑛 ​ ( 𝑚 ⋅ ∇ 𝑢 )

( 𝑚 ⋅ 𝑛 ) ​ Δ ​ 𝑢 .

Substituting these facts into (3.11) simplifies it to

𝐼 2

( 2 − 𝑛 2 ) ​ ∫ 0 𝑇 ∫ Ω | Δ ​ 𝑢 | 2 ​ d 𝑥 ​ d 𝑡 − 1 2 ​ ∫ 0 𝑇 ∫ ∂ Ω ( 𝑚 ⋅ 𝑛 ) ​ | Δ ​ 𝑢 | 2 ​ d 𝑆 ​ d 𝑡 .

(3.12)

Estimate of 𝐼 3 : A direct computation gives

𝐼 3

𝛾 2 ​ ∫ 0 𝑇 ∫ Ω 𝑚 ⋅ ∇ | ∂ 𝑡 𝑢 | 2 ​ d ​ 𝑥 ​ d ​ 𝑡

− 𝛾 ​ 𝑛 2 ​ ∫ 0 𝑇 ∫ Ω | ∂ 𝑡 𝑢 | 2 ​ d 𝑥 ​ d 𝑡 .

(3.13)

Combining (3.9)–(3.13) and using the star-shaped condition 𝑚 ⋅ 𝑛 ≥ 𝑐 0

0 , all volume integrals are bounded by ∫ 0 𝑇 𝐸 ​ ( 𝑡 ) ​ d 𝑡 or the initial energy.

We now establish the observability estimate.

Case 1: 𝛾

0 . In the undamped case the energy is conserved, which yields 𝐸 ​ ( 𝑡 )

𝐸 ​ ( 0 ) for all 𝑡 ∈ [ 0 , 𝑇 ] . A compactness-uniqueness argument provides a constant 𝐶 1

0 , independent of 𝛾 , such that

𝐸 ​ ( 0 ) ≤ 𝐶 1 ​ ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ d 𝑆 ​ d 𝑡 .

(3.14)

Case 2: 𝛾

0 . When damping is present, careful estimation of the volume terms in the identity above shows that they can be bounded by

1 𝛾 ​ 𝐸 ​ ( 0 ) + 𝛽 ​ ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ d 𝑆 ​ d 𝑡 ,

where 𝛽

0 is independent of 𝛾 . Together with the geometric control condition this yields

𝐸 ​ ( 0 ) ≤ 𝛽 ​ ( 1 + 𝛾 ) ​ ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ d 𝑆 ​ d 𝑡 .

(3.15)

Taking 𝐶 0

max ⁡ { 𝐶 1 , 𝛽 } (which is independent of 𝛾 ), we obtain for all 𝛾 ≥ 0

𝐸 ​ ( 0 ) ≤ 𝐶 0 ​ ( 1 + 𝛾 ) ​ ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ d 𝑆 ​ d 𝑡 .

(3.16)

This completes the proof of the observability inequality, with explicit 𝛾 -dependence through the factor ( 1 + 𝛾 ) and the main constant 𝐶 0 independent of 𝛾 .

∎

4.Proof of the Main Theorems

This section presents the proofs of the main stability results, Theorems 1.1 and 1.2. The core strategy involves analyzing the system satisfied by the difference of two solutions corresponding to different parameter sets . Let 𝑢 1 and 𝑢 2 be solutions of system (1.1) with parameters ( 𝜌 1 , 𝑓 1 , 𝑔 1 ) and ( 𝜌 2 , 𝑓 2 , 𝑔 2 ) , respectively, all belonging to the admissible set. Set

𝑢

𝑢 1 − 𝑢 2 , 𝜌

𝜌 1 − 𝜌 2 , 𝑓

𝑓 1 − 𝑓 2 𝑔

𝑔 1 − 𝑔 2 .

Then 𝑢 satisfies the inhomogeneous system

{ 𝜌 1 ​ ( 𝑥 ) ​ ∂ 𝑡 2 𝑢 + Δ 2 ​ 𝑢 + 𝛾 ​ ∂ 𝑡 𝑢

− 𝜌 ​ ( 𝑥 ) ​ ∂ 𝑡 2 𝑢 2 ,

( 𝑡 , 𝑥 ) ∈ ( 0 , 𝑇 ) × Ω ,

𝑢 ​ ( 0 , 𝑥 )

𝑓 ​ ( 𝑥 ) , ∂ 𝑡 𝑢 ​ ( 0 , 𝑥 )

𝑔 ​ ( 𝑥 ) ,

𝑥 ∈ Ω ,

𝑢 ​ ( 𝑡 , 𝑥 )

0 , ∂ 𝑢 ∂ 𝑛 ​ ( 𝑡 , 𝑥 )

0 ,

( 𝑡 , 𝑥 ) ∈ ( 0 , 𝑇 ) × ∂ Ω .

(4.1)

The source term − 𝜌 ​ ∂ 𝑡 2 𝑢 2 quantifies the impact of the discrepancy in the density coefficients.

4.1.Proof of Theorem 1.1 Proof.

We now establish a rigorous energy estimate for the difference solution 𝑢 in system (4.1). Let 𝑣

∂ 𝑡 𝑢 and define the energy functional

𝐸 ​ ( 𝑡 )

1 2 ​ ∫ Ω ( 𝜌 1 ​ ( 𝑥 ) ​ | 𝑣 ​ ( 𝑡 , 𝑥 ) | 2 + | Δ ​ 𝑢 ​ ( 𝑡 , 𝑥 ) | 2 ) ​ d 𝑥 .

To derive the energy evolution equation, multiply the PDE in (4.1) by 𝑣 and integrate over Ω

∫ Ω 𝜌 1 ​ ∂ 𝑡 2 𝑢 ​ 𝑣 ​ d ​ 𝑥 + ∫ Ω ( Δ 2 ​ 𝑢 ) ​ 𝑣 ​ d 𝑥 + 𝛾 ​ ∫ Ω | 𝑣 | 2 ​ d 𝑥

− ∫ Ω 𝜌 ​ ∂ 𝑡 2 𝑢 2 ​ 𝑣 ​ d ​ 𝑥 .

(4.2)

For the biharmonic term, since 𝑢 ∈ 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) and 𝑣 ∈ 𝐻 0 2 ​ ( Ω ) , two applications of Green’s identity gives

∫ Ω ( Δ 2 ​ 𝑢 ) ​ 𝑣 ​ d 𝑥

∫ Ω Δ ​ 𝑢 ​ Δ ​ 𝑣 ​ d 𝑥 ,

(4.3)

where the boundary terms vanish due to the conditions 𝑣 | ∂ Ω

0 and ∂ 𝑛 𝑣 | ∂ Ω

0 .

Substituting (4.3) into (4.2) and noting that ∂ 𝑡 2 𝑢

∂ 𝑡 𝑣 , we obtain

∫ Ω 𝜌 1 ​ ∂ 𝑡 𝑣 ​ 𝑣 ​ d ​ 𝑥 + ∫ Ω Δ ​ 𝑢 ​ Δ ​ 𝑣 ​ d 𝑥 + 𝛾 ​ ∫ Ω | 𝑣 | 2 ​ d 𝑥

− ∫ Ω 𝜌 ​ ∂ 𝑡 2 𝑢 2 ​ 𝑣 ​ d ​ 𝑥 .

Recognising the time derivatives of the energy components, this yields

d ​ 𝐸 d ​ 𝑡

− 𝛾 ​ ∫ Ω | 𝑣 | 2 ​ d 𝑥 − ∫ Ω 𝜌 ​ ∂ 𝑡 2 𝑢 2 ​ 𝑣 ​ d ​ 𝑥 .

We now estimate the source term. Using the Cauchy–Schwarz inequality and the assumption ‖ ∂ 𝑡 2 𝑢 2 ‖ 𝐿 ∞ ​ ( 0 , 𝑇 ; 𝐿 2 ​ ( Ω ) ) ≤ 𝑀 , we have

| ∫ Ω 𝜌 ​ ∂ 𝑡 2 𝑢 2 ​ 𝑣 ​ d ​ 𝑥 | ≤ ‖ 𝜌 ‖ 𝐿 ∞ ​ 𝑀 ​ ‖ 𝑣 ‖ 𝐿 2 ≤ ‖ 𝜌 ‖ 𝐿 ∞ ​ 𝑀 ​ 2 𝜌 min ​ 𝐸 ​ ( 𝑡 ) 1 / 2 .

Consequently, the energy dissipation inequality becomes

d ​ 𝐸 d ​ 𝑡 ≤ − 𝛾 ​ ‖ 𝑣 ‖ 𝐿 2 2 + ‖ 𝜌 ‖ 𝐿 ∞ ​ 𝑀 ​ 2 𝜌 min ​ 𝐸 ​ ( 𝑡 ) 1 / 2 .

(4.4)

Integrating (4.4) over [ 0 , 𝑇 ] gives

𝐸 ​ ( 𝑇 ) − 𝐸 ​ ( 0 ) ≤ − 𝛾 ​ ∫ 0 𝑇 ‖ 𝑣 ​ ( 𝑡 ) ‖ 𝐿 2 2 ​ d 𝑡 + ‖ 𝜌 ‖ 𝐿 ∞ ​ 𝑀 ​ 2 𝜌 min ​ ∫ 0 𝑇 𝐸 ​ ( 𝑡 ) 1 / 2 ​ d 𝑡 .

Since 𝐸 ​ ( 𝑇 ) ≥ 0 , we deduce

𝐸 ​ ( 0 ) ≤ 𝛾 ​ ∫ 0 𝑇 ‖ 𝑣 ​ ( 𝑡 ) ‖ 𝐿 2 2 ​ d 𝑡 + ‖ 𝜌 ‖ 𝐿 ∞ ​ 𝑀 ​ 2 𝜌 min ​ ∫ 0 𝑇 𝐸 ​ ( 𝑡 ) 1 / 2 ​ d 𝑡 + 𝐸 ​ ( 𝑇 ) .

(4.5)

To proceed, we need bounds on the integrals involving 𝐸 ​ ( 𝑡 ) . From the energy definition, we have ‖ 𝑣 ‖ 𝐿 2 2 ≤ 2 𝜌 min ​ 𝐸 ​ ( 𝑡 ) . Hence,

∫ 0 𝑇 ‖ 𝑣 ​ ( 𝑡 ) ‖ 𝐿 2 2 ​ 𝑑 𝑡 ≤ 2 𝜌 min ​ ∫ 0 𝑇 𝐸 ​ ( 𝑡 ) ​ 𝑑 𝑡 .

(4.6)

To bound 𝐸 ​ ( 𝑡 ) , we temporarily neglect the dissipation term in (4.4) to obtain

d ​ 𝐸 d ​ 𝑡 ≤ ‖ 𝜌 ‖ 𝐿 ∞ ​ 𝑀 ​ 2 𝜌 min ​ 𝐸 ​ ( 𝑡 ) 1 / 2 .

This inequality implies that

d d ​ 𝑡 ​ ( 𝐸 ​ ( 𝑡 ) 1 / 2 ) ≤ 1 2 ​ ‖ 𝜌 ‖ 𝐿 ∞ ​ 𝑀 ​ 2 𝜌 min ,

and integrating from 0 to 𝑡 yields

𝐸 ​ ( 𝑡 ) 1 / 2 ≤ 𝐸 ​ ( 0 ) 1 / 2 + 1 2 ​ ‖ 𝜌 ‖ 𝐿 ∞ ​ 𝑀 ​ 2 𝜌 min ​ 𝑡 .

Squaring both sides gives the uniform bound

𝐸 ​ ( 𝑡 ) ≤ 2 ​ 𝐸 ​ ( 0 ) + 𝐾 ​ ‖ 𝜌 ‖ 𝐿 ∞ 2 ​ 𝑇 2 , 𝐾

𝑀 2 2 ​ 𝜌 min .

(4.7)

From (4.7), we immediately obtain the integral estimates

∫ 0 𝑇 𝐸 ​ ( 𝑡 ) ​ 𝑑 𝑡 ≤ 𝑇 ​ ( 2 ​ 𝐸 ​ ( 0 ) + 𝐾 ​ ‖ 𝜌 ‖ 𝐿 ∞ 2 ​ 𝑇 2 ) ,

(4.8)

∫ 0 𝑇 𝐸 ​ ( 𝑡 ) 1 / 2 ​ 𝑑 𝑡 ≤ 𝑇 ​ 2 ​ 𝐸 ​ ( 0 ) + 𝐾 ​ ‖ 𝜌 ‖ 𝐿 ∞ 2 ​ 𝑇 2 ,

(4.9)

𝐸 ​ ( 𝑇 ) ≤ 2 ​ 𝐸 ​ ( 0 ) + 𝐾 ​ ‖ 𝜌 ‖ 𝐿 ∞ 2 ​ 𝑇 2 .

(4.10)

Substitution of (4.6) and (4.8)– (4.10) into (4.5) yields

𝐸 ​ ( 0 )

≤ 𝛾 ​ 2 𝜌 min ​ 𝑇 ​ ( 2 ​ 𝐸 ​ ( 0 ) + 𝐾 ​ ‖ 𝜌 ‖ 𝐿 ∞ 2 ​ 𝑇 2 )

• ‖ 𝜌 ‖ 𝐿 ∞ ​ 𝑀 ​ 2 𝜌 min ​ 𝑇 ​ 2 ​ 𝐸 ​ ( 0 )

• 𝐾 ​ ‖ 𝜌 ‖ 𝐿 ∞ 2 ​ 𝑇 2

• 2 ​ 𝐸 ​ ( 0 )

• 𝐾 ​ ‖ 𝜌 ‖ 𝐿 ∞ 2 ​ 𝑇 2 .

We now invoke the observability inequality (Theorem 3.3) for the solution 𝑢 of (4.1). By Theorem 3.3 there exists a constant 𝐶 0

𝐶 0 ​ ( Ω , 𝜌 , 𝑇 )

0 , independent of 𝛾 , such that

𝐸 ​ ( 0 ) ≤ 𝐶 0 ​ ( 1 + 𝛾 ) ​ ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ d 𝑆 ​ d 𝑡 .

(4.11)

It remains to connect ‖ 𝜌 ‖ 𝐿 ∞ directly with the boundary observation data. Consider the system at the initial time 𝑡

0 . From (4.1) we have

𝜌 1 ​ ∂ 𝑡 2 𝑢 ​ ( 0 ) + Δ 2 ​ 𝑓

− 𝜌 ​ ∂ 𝑡 2 𝑢 2 ​ ( 0 ) .

Taking the 𝐿 2 -norm and applying the triangle inequality gives

‖ Δ 2 ​ 𝑓 ‖ 𝐿 2 ≤ ‖ 𝜌 1 ‖ 𝐿 ∞ ​ ‖ ∂ 𝑡 2 𝑢 ​ ( 0 ) ‖ 𝐿 2 + ‖ 𝜌 ‖ 𝐿 ∞ ​ ‖ ∂ 𝑡 2 𝑢 2 ​ ( 0 ) ‖ 𝐿 2 .

(4.12)

To estimate ∂ 𝑡 2 𝑢 ​ ( 0 ) , we use the PDE at 𝑡

0 :

∂ 𝑡 2 𝑢 ​ ( 0 )

− 𝜌 1 − 1 ​ ( Δ 2 ​ 𝑓 + 𝛾 ​ 𝑔 + 𝜌 ​ ∂ 𝑡 2 𝑢 2 ​ ( 0 ) ) ,

which implies

‖ ∂ 𝑡 2 𝑢 ​ ( 0 ) ‖ 𝐿 2 ≤ 𝜌 min − 1 ​ ( ‖ Δ 2 ​ 𝑓 ‖ 𝐿 2 + 𝛾 ​ ‖ 𝑔 ‖ 𝐿 2 + ‖ 𝜌 ‖ 𝐿 ∞ ​ 𝑀 ) .

Inserting this into (4.12) and rearranging yields

‖ Δ 2 ​ 𝑓 ‖ 𝐿 2 ≤ 𝐶 1 ​ ( ‖ 𝑔 ‖ 𝐿 2 + ‖ 𝜌 ‖ 𝐿 ∞ ) ,

(4.13)

where 𝐶 1 depends on 𝜌 min , 𝜌 max , 𝑀 .

Recall that 𝐸 ​ ( 0 )

1 2 ​ ( ‖ 𝜌 1 ​ 𝑔 ‖ 𝐿 2 2 + ‖ Δ ​ 𝑓 ‖ 𝐿 2 2 ) . Using Poincaré’s inequality and elliptic estimates for the biharmonic operator, there exists a constant 𝑐 2

0 such that

𝐸 ​ ( 0 ) ≥ 𝑐 2 ​ ( ‖ 𝑔 ‖ 𝐿 2 2 + ‖ 𝑓 ‖ 𝐻 2 2 ) .

Combining this lower bound with (4.13) and the observability estimate (4.11), we obtain

‖ 𝜌 ‖ 𝐿 ∞ ≤ 𝐶 2 ​ 𝐸 ​ ( 0 ) 1 / 2 ≤ 𝐶 2 ​ 𝐶 0 ​ ( 1 + 𝛾 ) ​ ( ∫ 0 𝑇 ∫ ∂ Ω | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ​ d ​ 𝑆 ​ d ​ 𝑡 ) 1 / 2 ,

where 𝐶 2

𝐶 2 ​ ( 𝜌 min , 𝜌 max , 𝑀 , Ω ) .

Setting 𝐶

𝐶 2 ​ 𝐶 0 completes the proof of Theorem 1.1. ∎

4.2.Proof of Theorem 1.2 Proof.

We now prove the stability estimate for the initial displacement under the assumption 𝑔 1

𝑔 2 . Since 𝑔

0 , the initial energy simplifies to

𝐸 ​ ( 0 )

1 2 ​ ∫ Ω | Δ ​ 𝑓 ​ ( 𝑥 ) | 2 ​ d 𝑥

1 2 ​ ‖ Δ ​ 𝑓 ‖ 𝐿 2 ​ ( Ω ) 2 .

Applying the observability inequality (Theorem 3.3) yields

‖ Δ ​ 𝑓 ‖ 𝐿 ​ ² 2

2 ​ 𝐸 ​ ( 0 ) ≤ 2 ​ 𝐶 0 ​ ( 1 + 𝛾 ) ​ ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ 𝑑 𝑆 ​ 𝑑 𝑡 .

Since 𝑓

𝑓 1 − 𝑓 2 ∈ 𝐻 4 ​ ( Ω ) ∩ 𝐻 0 2 ​ ( Ω ) , elliptic regularity for the biharmonic operator gives a constant 𝐶 1

0 such that

‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) ≤ 𝐶 1 ​ ‖ Δ 2 ​ 𝑓 ‖ 𝐿 2 ​ ( Ω ) .

(4.14)

We now analyse system (4.1) at 𝑡

0 . Since 𝑔

0 , we have ∂ 𝑡 𝑢 ​ ( 0 )

0 . Evaluating the PDE at 𝑡

0 gives

𝜌 1 ​ ( 𝑥 ) ​ ∂ 𝑡 2 𝑢 ​ ( 0 , 𝑥 ) + Δ 2 ​ 𝑓 ​ ( 𝑥 )

− 𝜌 ​ ( 𝑥 ) ​ ∂ 𝑡 2 𝑢 2 ​ ( 0 , 𝑥 ) .

Taking the 𝐿 2 -norm of both sides yields

‖ Δ 2 ​ 𝑓 ‖ 𝐿 2 ≤ ‖ 𝜌 1 ‖ 𝐿 ∞ ​ ‖ ∂ 𝑡 2 𝑢 ​ ( 0 ) ‖ 𝐿 2 + ‖ 𝜌 ‖ 𝐿 ∞ ​ ‖ ∂ 𝑡 2 𝑢 2 ​ ( 0 ) ‖ 𝐿 2 .

(4.15)

By assumption, ‖ ∂ 𝑡 2 𝑢 2 ​ ( 0 ) ‖ 𝐿 2 ≤ 𝑀 . A standard computation using the original systems for 𝑢 1 and 𝑢 2 at 𝑡

0 shows there exists a constant 𝐶 3 , depending on 𝜌 min , 𝜌 max , 𝑀 , such that

‖ ∂ 𝑡 2 𝑢 ​ ( 0 ) ‖ 𝐿 2 ≤ 𝐶 3 ​ ( ‖ Δ 2 ​ 𝑓 ‖ 𝐿 2 + ‖ 𝜌 ‖ 𝐿 ∞ ) .

(4.16)

Substituting (4.16) into (4.15) and rearranging terms leads to

‖ Δ 2 ​ 𝑓 ‖ 𝐿 2 ≤ 𝐶 4 ​ ‖ 𝜌 ‖ 𝐿 ∞ ,

(4.17)

where 𝐶 4 depends only on 𝜌 min , 𝜌 max , 𝑀 .

Theorem 1.1 supplies the crucial link, estimating ‖ 𝜌 ‖ 𝐿 ∞ ​ ( Ω ) in terms of the boundary data:

‖ 𝜌 ‖ 𝐿 ∞ ​ ( Ω ) ≤ 𝐶 ​ ( 1 + 𝛾 ) 1 / 2 ​ ( ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ d 𝑆 ​ d 𝑡 ) 1 / 2 .

(4.18)

Finally, combining (4.14), (4.17) and (4.18) we obtain the desired stability estimate for the initial displacement:

‖ 𝑓 ‖ 𝐻 4 ​ ( Ω ) ≤ 𝐶 ​ ( 1 + 𝛾 ) 1 / 2 ​ ( ∫ 0 𝑇 ∫ ∂ Ω ( | ∂ Δ ​ 𝑢 ∂ 𝑛 | 2 + | Δ ​ 𝑢 | 2 ) ​ d 𝑆 ​ d 𝑡 ) 1 / 2 .

This completes the proof of Theorem 1.2. ∎

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