Inverse Problem for a Structural Acoustic System with Variable Coefficients

Abstract

We consider stability in an inverse problem of determining the material coefficient matrix for a coupled system that describes acoustic interactions, by the Riemannian geometrical approach. The stability is proved by the Carleman estimates and observability inequalities.

1 Introduction and Main Results

Let $\varOmega$ be a connected open bounded domain of $\textrm{IR}^3$ with boundary $\varGamma =\overline{{\varGamma _0} \cup {\varGamma _1}}$ and $\varGamma _0\cap \varGamma _1=\emptyset$, where $\varGamma _0$ and $\varGamma _1$ are open and nonempty. Moreover, $\varGamma _1$ is assumed to be convex and of class $C^2$, and $\varGamma _0\subset \textrm{IR}^2$ to be flat with smooth boundary $\partial \varGamma _0$. For a possible geometric graphics of the structural acoustic chamber $\varOmega$, we refer to [23, 25].

We consider the following coupling system on the finite time interval (0, T):

$$\begin{aligned} \left\{ \begin{array}{ll} z_{tt}-{{\mathcal {A}}}z=0 &{}\quad {{\textrm{in}}} \ \varOmega \times (0,T),\\ \frac{\partial z}{\partial \nu _{\mathcal {A}}}=0 &{}\quad {{\textrm{on}}}\ \varGamma _1\times (0,T),\\ \frac{\partial z}{\partial \nu _{{\mathcal {A}}}}=v_t &{}\quad {{\textrm{on}}}\ \varGamma _0\times (0,T),\\ v_{tt}+{\mathcal {A}}_0^2v=-z_t &{}\quad {{\textrm{on}}} \ \varGamma _0\times (0,T),\\ v=\frac{\partial v}{\partial n_0}=0 &{}\quad {{\textrm{on}}}\ \partial \varGamma _0 \times (0,T) ,\\ (z(x,0),z_t(x,0))=(z_0,0) &{}\quad {{\textrm{in}}}\ \varOmega ,\\ (v(x,0),v_t(x,0))=(v_0,0) &{}\quad {{\textrm{on}}} \ \varGamma _0, \end{array} \right. \end{aligned}$$

(1.1)

where ${\mathcal {A}}z={{\textrm{div}}}A(x)\nabla z$ and ${\mathcal {A}}_0v={{\mathrm{div_{\varGamma _0}}}}A_0(x)\nabla v.$ In (1.1), A(x) and $A_0(x)$ are symmetric, positive matrices satisfying

$$\begin{aligned} A(x)=A_0(x){\quad \text{ for }\quad }x\in \varGamma _0. \end{aligned}$$

Moreover, in (1.1), z denotes the acoustic velocity potential in $\varOmega$, which is a wave-type equation with the Neumann boundary condition and v describes the vertical displacement of the flat $\varGamma _0$ . In addition, $\nu ,$ $n_0,$ ${{\textrm{div}}},$ and $\nabla$ are the outward unit normal vector of $\varOmega$ along $\varGamma ,$ the outward unit normal vector of $\varGamma _0$ along $\partial \varGamma _0,$ the divergence, and the gradient, respectively, in the Euclidean metric. Finally, $\frac{\partial z}{\partial \nu _{\mathcal {A}}}=\left\langle \nabla z, A(x)\nu \right\rangle$ and $\frac{\partial v}{\partial n_0}=\left\langle \nabla v,A_0(x)n\right\rangle .$

We assume that the matrix $A_0(x)$ is given but the matrix A(x) is unknown which needs to be determined. Note that the map $A(x)\rightarrow \{z,v\}$ is nonlinear. Thus the inverse map $\{z,v\}\rightarrow A(x)$ is also nonlinear. We have taken the initial data $z_t(x,0)=v_t(x,0)=0$ in order to make the even extensions of the solutions z and v to $\varOmega \times [-T,T].$ The extended solutions retain the same regularity in the domain $\varOmega \times [-T,T].$ The explicit regularity needed in our inverse problems will be specified in Sect. 2. Therefore, here and after, we consider all the PDE systems in the domain $Q=\varOmega \times [-T,T]$ with the lateral boundary $\varSigma =\varGamma \times [-T,T].$

As for the nonlinear inverse problem $\left\{ z,v \right\} \rightarrow A(x)$ of system (1.1), we view $z_0$ and $v_0$ as the input, and the acceleration of the elastic plate $v_{tt}|_{\varSigma _0}$, a physically measurable quantity, as the output (observation). More precisely, we consider the following inverse problem:

• Uniqueness of the inverse problem for system (1.1)

Can the principal coefficients matrix A(x) be uniquely determined by the acceleration of the elastic plate $v_{tt}|_{\varSigma _0}$ by finite many times changing initial values suitably? In other words, do finitely many $v_{tt}|_{\varSigma _0}=0$ imply $A_1(x)=A_2(x)$, a.e. $x\in \varOmega$?

• Stability of the inverse problem for system (1.1)

For a matrix $A(x)=(a_{ij}(x))_{1\le i,j\le 3},$ we define the following norm:

$$\begin{aligned} ||A||^2_{H^1(\varOmega )}=\sum _{i,j=1}^3||a_{ij}(x)||^2_{H^1(\varOmega )}. \end{aligned}$$

Is it possible to estimate $||A_1-A_2||_{H^1(\varOmega )}$ by some suitable norms of the difference of the corresponding plate accelerations $(v_{2k}-v_{1k})_{tt}|_{\varSigma _0}$?

For our purposes, we shall first consider the linearized inverse problems in the following setting. Let

$$\begin{aligned} z_{ik}(x,t)=z(A_i(x),a_k)\quad \text{ and }\quad v_{ik}(x,t)=v_{ik}(A_i(x),a_k), \end{aligned}$$

respectively, solve (1.1) with respect to the coefficient matrices $A_i(x)$ and the initial values

$$\begin{aligned} {[}z_{ik}(x,0),\partial _tz_{ik}(x,0);v_{ik}(x,0),\partial _tv_{ik}(x,0)]=[a_k,0;v_0,0], \end{aligned}$$

where $v_0$ is a fixed function, for $1\le i\le 2$ and $1\le k\le 9.$ Denote

$$\begin{aligned}{} & {} B(x)=(b_{ij})_{3\times 3}=A_2(x)-A_1(x), \quad w_k(x,t)=z_{2k}(x,t)-z_{1k}(x,t),\\{} & {} R_k(x,t)=z_{2k}(x,t)\ \ {{\textrm{in}}}\ Q,\quad \text{ and }\quad u_k(x,t)=v_{2k}(x,t)-v_{1k}(x,t)\ \ {{\textrm{in}}}\ \varSigma _0. \end{aligned}$$

For the sake of simplicity, for $i=1,2$, we denote

$$\begin{aligned}{} & {} Z_i(x,t)=(z_{i1},\ldots , z_{i9})^\textrm{T},\quad V_i(x,t)=(v_{i1},\ldots , v_{i9})^\textrm{T},\\{} & {} Z_0(x,t)=(a_1,\ldots ,a_9)^\textrm{T},\quad V_0(x,t)=(v_0,\ldots ,v_0)^\textrm{T},\\{} & {} W(x,t)=(w_1,\ldots ,w_9)^\textrm{T},\quad U(x,t)=(u_1,\ldots ,u_9)^\textrm{T}, \quad \text{ and }\\{} & {} R(x,t)=(z_{21},\ldots ,z_{29})^\textrm{T}, \end{aligned}$$

where the superscript T denotes the transpose. Moreover, we let ${\mathcal {A}}_i={{\textrm{div}}}A_i(x)\nabla$ and $\frac{\partial z}{\partial \nu _{{\mathcal {A}}_i}}=\left\langle A_i(x)\nabla z,\nu \right\rangle$ for $i=1, 2$. Clearly, the couple $\left\{ W,U \right\}$ satisfies the following system.

$$\begin{aligned} \left\{ \begin{array}{ll} W_{tt}-{\,{{\textrm{div }}}\,}A_1(x)\nabla W=\,{{\textrm{div }}}\,B(x)\nabla R(x,t)&{}\quad {\quad \text{ in }\quad }\varOmega \times (-T,T),\\ \frac{\partial W}{\partial \nu _{{\mathcal {A}}_1}}=0&{}\quad \quad \text{ on }\quad \varGamma _1\times (-T,T),\\ \frac{\partial W}{\partial \nu _{{\mathcal {A}}_1}}=U_t &{}\quad \quad \text{ on }\quad \varGamma _0\times (-T,T),\\ U_{tt}+{\mathcal {A}}^2_0U=-W_t&{}\quad \quad \text{ on }\quad \varGamma _0\times (-T,T),\\ U=\frac{\partial U}{\partial n_0}=0 &{}\quad \quad \text{ on }\quad \partial \varGamma _0 \times (-T,T) ,\\ W(x,0)=W_t(x,0)=0 &{}\quad {\quad \text{ in }\quad }\varOmega ,\\ U(x,0)=U_t(x,0)=0 &{}\quad \quad \text{ on }\quad \varGamma _0, \end{array} \right. \end{aligned}$$

(1.2)

where ${\,{{\textrm{div }}}\,}A_1\nabla W=({\,{{\textrm{div }}}\,}A_1\nabla w_1,\ldots ,{\,{{\textrm{div }}}\,}A_1\nabla w_9)\textrm{T}.$

We introduce

$$\begin{aligned} g=A^{-1}(x)\quad {{\textrm{for }}}\ \ x\in \textrm{IR}^3, \end{aligned}$$

as a Riemannian metric on $\textrm{IR}^3$ and consider $(\textrm{IR}^3, g)$ as a Riemannian manifold. Let

$$\begin{aligned} g(X,Y)=\left\langle X,Y \right\rangle _g =\left\langle A^{-1}(x)X,\ Y \right\rangle \quad {{\textrm{for}}} \ X,Y\in \textrm{IR}^3_x,\quad x\in \textrm{IR}^3, \end{aligned}$$

where $\left\langle \cdot ,\cdot \right\rangle$ denotes the Euclidean product of $\textrm{IR}^3$. Let D be the Levi–Civita connection in the metric g, and we have

$$\begin{aligned} DH(X,Y)=\left\langle D_YH,X \right\rangle _g\quad {{\textrm{for}}}\ \ X,\ Y,\ H\in \textrm{IR}^3_x,\quad \ x\in \varOmega . \end{aligned}$$

(1.3)

We need the following main assumptions.

Assumption (A.1) on the metric $g=A^{-1}(x)$: Assume that there exists a strictly convex function $\upsilon :{\overline{\varOmega }}\rightarrow (0,+\infty )$ of class $C^3,$ such that the following three properties hold.

• (i) $\left. \frac{\partial \upsilon }{\partial \nu _{{\mathcal {A}}}} \right| _{\varGamma _1}=0$;

• (ii) There exists a positive constant $\alpha >0$, such that

  $$\begin{aligned} D^2\upsilon (X,X)\ge \alpha |X|_g^2,\quad \forall X\in \textrm{IR}^3_x,\quad \forall x\in {\overline{\varOmega }}, \end{aligned}$$

  where D is the connection of the metric $g=A^{-1}(x);$

• (iii) $\upsilon (x)$ has no critical point on ${\overline{\varOmega }}$, namely,

  $$\begin{aligned} \mathop {\inf }\limits _{x \in \varOmega } |\nabla _g \upsilon |_g\ge \beta >0. \end{aligned}$$

In the case of constant coefficients, conditions (i) and (ii) in (A.1) are due to the Neumann boundary conditions which are the physically correct boundary conditions of the hyperbolic problem and introduced in [27, Sect. 5]. We mention that in [28, Appendix B], the authors have given some constructions of functions satisfying condition (i). Condition (iii) is needed for the validity of the pointwise Carleman estimate. Condition (ii) means that v is an escape function which depends on the curvature of the metric $g=A^{-1}(x).$ For the case of constant coefficients, $\upsilon (x) = |x-x_0|^2$ is one of the choices, where $x_0$ is a fixed point outside ${\overline{\varOmega }}.$ For the general cases, there are some examples in [34, Chap. 2] to show how to find an escape function. We here given an example.

An example satisfying conditions (i) and (ii) in assumption (A.1). Similar to [25, Example 2.1], for a given

$$\begin{aligned} A(x)=\left( {\begin{array}{*{20}{c}} \frac{1}{4}(1+|x|^2)^2&{} \ 0&{} 0\\ 0&{} \ \frac{1}{4}(1+|x|^2)^2&{} 0\\ 0&{} \ 0&{} \frac{1}{4}(1+|x|^2)^2 \end{array}} \right) \quad {{\textrm{for}}}\ x=(x_1,x_2,x_3)\in \textrm{IR}^3, \end{aligned}$$

the metric g(x) is

$$\begin{aligned} g(x)=A^{-1}(x)=\left( {\begin{array}{*{20}{c}} \frac{4}{(1+|x|^2)^2}&{} \ 0&{} 0\\ 0&{} \ \frac{4}{(1+|x|^2)^2}&{} 0\\ 0&{} \ 0&{} \frac{4}{(1+|x|^2)^2} \end{array}} \right) . \end{aligned}$$

Let

$$\begin{aligned} {\mathcal {M}}=\{(x_1,x_2,x_3,x_4)\in \textrm{IR}^4:\ x_1^2+x_2^2+x_3^2+(x_4-1)^2=1\}, \end{aligned}$$

a sphere of $\textrm{IR}^4$ with radius 1. Let $p=(0,0,0,2).$ We define the stereographic projection P as

$$\begin{aligned} P: {\mathcal {M}}\backslash p\rightarrow (\textrm{IR}^3,g),\quad \!\!\!\! P(x)=\frac{1}{2-x_4}(x_1,x_2,x_3)\!\!{\quad \text{ for }\quad }\!\! x=(x_1,x_2,x_3,x_4)\in {\mathcal {M}}\backslash p. \end{aligned}$$

Then P is an isometry, which implies that the curvature of $(\textrm{IR}^3,g)$ is 1.

Let $p_0=(1,0,0,1)\in {\mathcal {M}}.$ Denote

$$\begin{aligned} {\mathcal {C}}(r,\theta _1,\theta _2)= & {} (\cos r,\sin r\cos \theta _1,\sin r\sin \theta _1\cos \theta _2,1+\sin r\sin \theta _1\sin \theta _2)\ {\quad \text{ for }\quad }\\{} & {} \quad 0<r<\frac{\pi }{2}. \end{aligned}$$

Then

$$\begin{aligned} B_{{\mathcal {M}}}(p_0,r_0)=\{{\mathcal {C}}(r,\theta _1,\theta _2):\ 0\le r\le r_0, 0\le \theta _1\le \pi , 0\le \theta _2<2\pi \} \end{aligned}$$

is a geodesic ball of ${\mathcal {M}}$ centered at $p_0$ with radius $r_0\in \left( 0,\frac{\pi }{2}\right) .$ Let $x_0=P(p_0)=(1,0,0)\in \textrm{IR}^3.$ Then, the geodesic ball of $(\textrm{IR}^3, g)$ centered at $x_0$ with radius $0<r_0<\frac{\pi }{2}$ is given by

$$\begin{aligned} B_g(x_0,r_0)= & {} P(B_{{\mathcal {M}}}(p_0,r_0))\\= & {} \Big \{\frac{1}{1-\sin r\sin \theta _1\sin \theta _2}(\cos r,\ \sin r\cos \theta _1,\ \sin r\sin \theta _1\cos \theta _2):\\{} & {} \quad 0\le r\le r_0, 0\le \theta _1\le \pi , 0\le \theta _2<2\pi \Big \}. \end{aligned}$$

Let $\rho (x)=d_g(x, x_0)$ be the distance function subject to metric g from $x\in \textrm{IR}^3$ to $x_0.$ Let $H=\frac{1}{2} D\rho ^2.$ Then by [34, Theorem 2.5], condition (i) holds for the escape function $\upsilon (x)=\rho ^2(x),$ provided that $\varOmega \subset B_g(x_0,r_0).$

Based on the above discussions, we give the following example satisfying condition (ii).

Example 1.1

Let $0<r_0<\frac{\pi }{2}.$ Set

$$\begin{aligned} \varOmega =(B_g(x_0,r_0)\cap \{(x_1,0,x_3)\in \textrm{IR}^3\})\backslash \{x\in \textrm{IR}^3:\ |x|>1\}, \end{aligned}$$

$$\begin{aligned} \varGamma _1= & {} \Big \{\frac{1}{1-\sin r\sin \theta _2}(\cos r,0,\sin r\cos \theta _2):\nonumber \\{} & {} \quad -r_0<r<r_0,0\le \theta _2<2\pi \}\cap \{x\in \textrm{IR}^3:|x|\le 1\Big \},\end{aligned}$$

(1.4)

$$\begin{aligned} \varGamma _0= & {} \Big \{\frac{1}{1-\sin r_0\sin \theta _1\sin \theta _2} (\cos r_0,\ \sin r_0\cos \theta _1,\ \sin r_0\sin \theta _1\cos \theta _2): \nonumber \\{} & {} \quad 0\le \theta _1\le \pi ,0\le \theta _2\le \pi \Big \}. \end{aligned}$$

(1.5)

Then conditions (i) and (ii) hold for the triple $\{(\varOmega ,g),\varGamma _0,\varGamma _1\}.$

It remains to show that the above example meets condition (i). Indeed, for a fixed $\theta _2\in [0,2\pi ),$ it is easy to see that $\gamma (r)=(\cos r,\sin r\cos \theta _1,\sin r\cos \theta _2,1+\sin r\sin \theta _2)$ is a normal geodesic of ${\mathcal {M}}$ through $\gamma (0)=p_0=(1,0,0,1).$ Then

$$\begin{aligned} \beta (r)=P(\gamma (r))=\frac{1}{1-\sin r\sin \theta _2}(\cos r,\sin r\cos \theta _1,\sin r\cos \theta _2) \end{aligned}$$

is a normal geodesic of $(\textrm{IR}^3,g)$ through $\beta (0)=x_0=(1,0,0).$ Notice that $\rho (\beta (r))=r.$ Then

$$\begin{aligned}{} & {} D\rho (\beta (r))\\{} & {} \quad =\frac{1}{(1-\sin r\sin \theta _2)^2}(-\sin r+\sin \theta _1\sin \theta _2,\,\cos r\cos \theta _1,\ \cos r\sin \theta _1\cos \theta _2). \end{aligned}$$

Let

$$\begin{aligned} u_1(r,\theta _2)=\frac{\cos r}{1-\sin r\sin \theta _2},\quad u_2(r,\theta _2)=0,\quad u_3(r,\theta _2)=\frac{\sin r\cos \theta _2}{1-\sin r\sin \theta _2}. \end{aligned}$$

Then

$$\begin{aligned} \nu |_{\varGamma _1}=\frac{(u_{1r},0,u_{3r})\times (u_{1\theta _2},0,u_{3\theta _2})}{|(u_{1r},0,u_{3r})\times (u_{1\theta _2},0,u_{3\theta _2})|}\sim (0,-1,0), \end{aligned}$$

which implies that ${\langle }D\rho ,\nu {\rangle }|_{\varGamma _1}={\langle }D\rho ,\nu {\rangle }|_{\theta _1=\frac{\pi }{2}}=0.$ Thus, condition (i) follows.

For given constants $0<c<1$ and $\,{{\mathrm{\gamma }}}\,>0$ small, we fix $T>0$ satisfying

$$\begin{aligned} \,{{\mathrm{\gamma }}}\,\left[ \max _{x\in \varOmega }\upsilon (x)-cT^2\right] <\log \min _{x\in \varOmega }\textrm{e}^{\,{{\mathrm{\gamma }}}\,\upsilon (x)}. \end{aligned}$$

(1.6)

Let $a(x)=(a_1(x),\ldots ,a_9(x))\textrm{T}.$ We set

$$\begin{aligned} G(x)= & {} \Big (a_{x_1x_1}(x),a_{x_1x_2}(x),a_{x_2x_3}(x),a_{x_2x_2}(x), a_{x_2x_3}(x),a_{x_3x_3}(x),\\{} & {} \times a_{x_1}(x),a_{x_2}(x),a_{x_3}(x)\Big ) \end{aligned}$$

for $x\in \varOmega .$ Note that G(x) has 81 components, and is a $9\times 9$ matrix of functions.

We further make the following assumption.

Assumption (A.2) Functions $a_1, \ldots , a_9$ are given such that

$$\begin{aligned} \det G(x)\not =0{\quad \text{ for }\quad }x\in \varOmega . \end{aligned}$$

We mention that such an example has been given in [9].

Moreover, for a given positive constant $C_0$, we denote an admissible set of A(x) as

$$\begin{aligned} {\mathcal {U}}(C_0)=\left\{ A\in C^5({\overline{\varOmega }},\textrm{IR}^{3\times 3})\ |\,A(x)=A_0(x)\ x\in \varGamma _0;\,\, ||A||_{C^5({\overline{\varOmega }})}\le C_0\right\} .\nonumber \\ \end{aligned}$$

(1.7)

Our main results are the following.

Theorem 1.1

(Uniqueness of the inverse problem) Let the assumption (A.1) of the metric $g=A_1^{-1}(x)$ and assumption (A.2) hold. Let T satisfy (1.6). Assume that $A_1(x),$ $A_2(x)\in {\mathcal {U}}(C_0),$ $v_0\in H_0^2(\varGamma _0),$ and $a(x)\in H^4(\varOmega ),$ such that

$$\begin{aligned} ||\partial _t^2R||_{L^{\infty }(-T,T;W^{2,\infty }(\varOmega ))} +||\partial _t^3R||_{L^{\infty }(-T,T;W^{2,\infty }(\varOmega ))}\le M_0<+\infty . \end{aligned}$$

(1.8)

Then $\partial _t^2U|_{\varSigma _0}=0$ implies that $B(x)=0$ for $x\in \varOmega .$

Theorem 1.2

(Stability of the inverse problem) Let all the assumptions in Theorem 1.1 hold. Let $A_1(x),$ $A_2(x)\in {\mathcal {U}}(C_0).$ Let $(a_j,0,v_0,0)\in {\mathcal {F}}$ for $1\le j\le 9,$ where ${\mathcal {F}}$ is given by (2.11). Then there exists a positive constant $C=C(T,\varOmega ,\varGamma _0,C_0,M_0,a,v_0)$ such that

$$\begin{aligned} ||B(x)||^2_{H^1(\varOmega )}\le C\left( ||\partial _t^2U_{tt}||^2_{L^2(\varSigma _0)}+||{\mathcal {A}}_0^2 U_{tt}||^2_{L^2(\varSigma _0)}\right) . \end{aligned}$$

(1.9)

The PDE system (1.1) describing acoustic interactions has been known and studied for some time (e.g., see [6, 7]). Physical motivation for studying this kind of problem comes from a variety of engineering applications that arise, for example, in the context of controlling the pressure in a helicopter’s cabin or reducing unwanted cabin noise generated by some exterior field. In the case where $A(x)=I_3$ the $3\times 3$ identity matrix, many papers contributed to various topics: stability, controllability, regularity, and inverse problems [1,2,3,4, 23]. In [23], an inverse problem of

$$\begin{aligned} \left\{ \begin{array}{ll} z_{tt}-\Delta z=qz &{}\quad {{\textrm{in}}} \ \varOmega \times (0,T),\\ \frac{\partial z}{\partial \nu _{\mathcal {A}}}=0 &{}\quad {{\textrm{on}}}\ \varGamma _1\times (0,T),\\ \frac{\partial z}{\partial \nu _{{\mathcal {A}}}}=v_t &{}\quad {{\textrm{on}}}\ \varGamma _0\times (0,T),\\ v_{tt}+\varDelta _\varGamma ^2v+\varDelta _\varGamma ^2v_t=-z_t &{}\quad {{\textrm{on}}} \ \varGamma _0\times (0,T),\\ v=\frac{\partial v}{\partial n_0}=0 &{}\quad {{\textrm{on}}}\ \partial \varGamma _0 \times (0,T) ,\\ (z(x,0),z_t(x,0))=(z_0,z_1) &{}\quad {{\textrm{in}}}\ \varOmega ,\\ (v(x,0),v_t(x,0))=(v_0,v_1) &{}\quad {{\textrm{on}}} \ \varGamma _0, \end{array} \right. \end{aligned}$$

was studied, where only the stability about q was obtained.

In the case where A(x) is not a constant matrix, not much literature (e.g., [25, 33]) is known for such a case. To the best knowledge of the authors, the present paper for the first time establishes the uniqueness and the Lipschitz stability (Theorems 1.1 and 1.2) in determining the important material coefficients matrix A(x) of system (1.1) with the finitely many observation data $v_{tt}.$ Moreover, the assumption (A.1) of the metric $g=A^{-1}(x)$ plays a key role to guarantee that the interior information of solutions to the system arrives at boundary $\varGamma _0$.

Inverse problems of PDEs have been the object of numerous studies not only at the theoretical level but also the practical. It is known that the Carleman estimates and microlocal analysis play an essential role in the inverse problems. We refer to [5, 8,9,10,11,12,13,14,15,16,17, 21, 26, 29,30,31,32, 35] and the references therein. Here, we shall adopt a differential geometrical approach [34] to study the inverse problems of system (1.1).

The rest of this paper is organized as follows: In Sect. 2, we give the Carleman estimates and observability inequalities for problem (1.1). Section 3 focuses on the proofs of Theorems 1.1 and 1.2. Some concluding remarks are given in the last Sect. 4.

2 Some Key Lemmas and Theorems

We introduce an abstract operator-theoretic formulation associated with (1.1) as in [23]. To achieve this, we consider an operator on $L^2(\varOmega )$ as follows.

$$\begin{aligned} {\mathcal {A}}u=\,{{\textrm{div }}}\,A(x)\nabla u\quad {{\textrm{with}}}\quad D({\mathcal {A}})=\left\{ \ z\in H^2(\varOmega ): {\frac{\partial z}{\partial \nu _{{\mathcal {A}}}}} |_{\varGamma }=0\right\} . \end{aligned}$$

(2.1)

It is easy to check that $-{\mathcal {A}}$ is a nonnegative, self-adjoint operator. We define the Neumann map $z=Np:$ $L^2(\varGamma _0)\rightarrow L^2(\varOmega )$ by:

$$\begin{aligned} \left\{ \begin{array}{ll} \,{{\textrm{div }}}\,A(x)\nabla (Np)=0&{}\quad {{\textrm{in}}}\ \varOmega ,\\ \frac{\partial Np}{\partial \nu _{{\mathcal {A}}}}=0&{}\quad \text{ on }\quad \varGamma _1,\\ \frac{\partial Np}{\partial \nu _{{\mathcal {A}}}}=p&{}\quad \text{ on }\quad \varGamma _0. \end{array} \right. \end{aligned}$$

(2.2)

It is well known that $N\in {\mathcal {L}}(L^2(\varGamma _0), H^{3/2}(\varOmega ))$ by the elliptic theory (see [22]). Then, by the Green’s formula and [18, 23], the operator $-N^*{\mathcal {A}}$ have the following property

$$\begin{aligned} -N^{*}{\mathcal {A}}\zeta = \left\{ \begin{array}{ll} \zeta &{}\quad {{\textrm{on}}}\ \varGamma _0\\ 0 &{}\quad \text{ on }\quad \varGamma _1 \end{array} \right. {\quad \text{ for }\quad }\zeta \in D({\mathcal {A}}). \end{aligned}$$

(2.3)

Extending $\zeta \in D({\mathcal {A}})$ by continuity to $\zeta \in H^1(\varOmega ).$ We set

$$\begin{aligned} {\mathcal {B}}=-{\mathcal {A}}N:\ L^2(\varGamma _0)\rightarrow [H^1(\varOmega )]^{\prime }, \end{aligned}$$

(2.4)

where $[H^1(\varOmega )]^{\prime }$ is the dual of $H^1(\varOmega )$ related to $L^2(\varOmega ).$ Then ${\mathcal {B}}^{*}=-N^{*}{\mathcal {A}}.$ Therefore, by (2.3), ${\mathcal {B}}^{*}$ is the restriction of the trace map from $H^1(\varOmega )$ to $H^{\frac{1}{2}}(\varGamma _0).$

Let ${{{\mathcal {A}}}}_0 v=\,{{\textrm{div }}}\,_{\varGamma _0}A_0(x)\nabla v$ be given in (1.2). Define

$$\begin{aligned} {\mathcal {C}}={\mathcal {A}}^2_0:\ D({\mathcal {C}})\rightarrow L^2(\varGamma _0), \quad D({\mathcal {C}})=\left\{ v\in H_0^2(\varGamma _0): {\mathcal {A}}^2_0v\in L^2(\varGamma _0) \right\} , \end{aligned}$$

(2.5)

where

$$\begin{aligned} H_0^2(\varGamma _0)=\left\{ v\in H^2(\varGamma _0): v\left| _{\partial \varGamma _0}={\frac{\partial v}{\partial n_0}}\right| _{\partial \varGamma _0}=0\right\} . \end{aligned}$$

It is easy to check that ${\mathcal {C}}$ is self-adjoint, positive definite and

$$\begin{aligned} D\left( {\mathcal {C}}^{\frac{1}{2}}\right) =H_0^2(\varGamma _0). \end{aligned}$$

Based on the original system and the above setting, we set

$$\begin{aligned} \varLambda =\left( {\begin{array}{*{20}{c}} 0&{}I&{}0&{}0\\ {\mathcal {A}}&{}0&{}0&{}{\mathcal {B}}\\ 0&{}0&{}0&{}I\\ 0&{}-{\mathcal {B}}^{*}&{}-{\mathcal {C}}&{}0 \end{array}} \right) :\ D(\varLambda )\subset \varXi \rightarrow \varXi , \end{aligned}$$

(2.6)

where

$$\begin{aligned} \varXi =H^1(\varOmega )\times L^2(\varOmega )\times H_0^2(\varGamma _0)\times L^2(\varGamma _0). \end{aligned}$$

(2.7)

The domain of $\varLambda$ is given by

$$\begin{aligned} D(\varLambda )= & {} \left\{ (z_0,z_1,v_0,v_1)^\textrm{T}:z_0\in H^2(\varOmega ), z_1\in H^1(\varOmega ), v_1\in H_0^2(\varGamma _0), \right. \nonumber \\{} & {} \left. \frac{\partial z_0}{\partial \nu _{{\mathcal {A}}}}=v_1\ {{\textrm{on}}}\ \varGamma _0,v_0\in D({\mathcal {C}}) \right\} . \end{aligned}$$

(2.8)

Then the original system can be re-written as the following abstract evolution equation

$$\begin{aligned} \frac{\textrm{d}{\mathcal {E}}}{\textrm{d}t}=\varLambda {\mathcal {E}},\quad {\mathcal {E}}(x,0)={\mathcal {E}}_0, \end{aligned}$$

(2.9)

where ${\mathcal {E}}=(z,z_t,v,v_t)^\textrm{T}$ and ${\mathcal {E}}_0=(z_0,z_1,v_0,v_1)^\textrm{T}.$ Therefore, the semigroup theory (e.g., see [1, 2]) yields that $\varLambda$ is the generator of a $C_0$-semigroup on $\varXi$, and

$$\begin{aligned} \begin{array}{l} {\mathcal {E}}_0\in \varXi \quad {{\textrm{implies}}}\ \left\{ z,z_t,v, v_t \right\} \in C(0,T;\varXi ),\\ {\mathcal {E}}_0\in D(\varLambda ) \quad {{\textrm{implies}}}\ \left\{ z,z_t,v, v_t \right\} \in C(0,T;D(\varLambda )). \end{array} \end{aligned}$$

(2.10)

Moreover, the time interval of (2.10) can be evenly extended to $[-T,T]$. For our inverse problem, we let

$$\begin{aligned}{} & {} {\mathcal {F}}=D(\varLambda ^6)\cap \left\{ [z_0,z_1,v_0,v_1]^\textrm{T}:z_0\in H^7(\varOmega ), z_1\in H^6(\varOmega ),\right. \nonumber \\{} & {} \quad v_0\in H_0^2(\varGamma _0)\cap H^7(\varGamma _0),v_1\in H_0^2(\varGamma _0)\cap H^6(\varGamma _0) \rbrace . \end{aligned}$$

(2.11)

Similarly, we have the abstract equation for the linearized system (1.2):

(2.12)

where $F(x,t)=\,{{\textrm{div }}}\,B(x)\nabla R(x,t).$

2.1 A Carleman Estimate

There are many papers on the Carleman estimates for the wave equation, for example, [10, 28] and the references therein. Among them, there is a compact form of such Carleman estimates from [10].

Suppose that $\upsilon :{\overline{\varOmega }}\rightarrow (0,+\infty )$ is a strictly convex function that satisfies the assumption (A.1) in the metric $g=A^{-1}(x).$ Let

$$\begin{aligned} \psi (x,t)= & {} \upsilon (x)-ct^2{\quad \text{ for }\quad }x\in \varOmega ,\quad t\in [-T,T], \end{aligned}$$

(2.13)

$$\begin{aligned} \varphi (x,t)= & {} \textrm{e}^{\gamma \psi },\quad (x,t)\in Q, \end{aligned}$$

(2.14)

where $\gamma >0$ is a constant. Set

$$\begin{aligned} m=\mathop {\min }\limits _{x \in {\overline{\varOmega }}}\upsilon (x),\quad d=\min _{x\in \overline{\varOmega }}\varphi (x,0)\ge \textrm{e}^{\gamma m}. \end{aligned}$$

Suitably choose $0<c<1$ and $T>0$, such that

$$\begin{aligned} \gamma \mathop {\max }\limits _{x \in {\overline{\varOmega }}}\upsilon (x)<\log d+c\gamma T^2. \end{aligned}$$

(2.15)

Then $\varphi$ satisfies

$$\begin{aligned} \varphi (x,0)\ge d,\quad \varphi (x,T)=\varphi (x,-T)<d\quad {{\mathrm{uniformly\ on}}}\ {\overline{\varOmega }}. \end{aligned}$$

(2.16)

Therefore, for given $\varepsilon >0$ small, we choose $\delta >0$ such that

$$\begin{aligned} \varphi (x,t)\ge & {} d-\varepsilon {\quad \text{ for }\quad }(x,t)\in {\overline{\varOmega }}\times [-\delta ,\delta ],\end{aligned}$$

(2.17)

$$\begin{aligned} \varphi (x,t)\le & {} d-2\varepsilon {\quad \text{ for }\quad }(x,t)\in {\overline{\varOmega }}\times ([-T,-T+2\delta ]\cup [T-2\delta ,T]).\nonumber \\ \end{aligned}$$

(2.18)

Let $P{v}=\partial _t^2{v}-{{\textrm{div}}}A(x)\nabla {v}$ and

$$\begin{aligned} {{{\mathcal {H}}}}= & {} \Bigg \lbrace {v}\in H^1(-T,T;L^2(\varOmega ))\cap L^2(-T,T;H^1(\varOmega )):\\{} & {} \quad \,\, \partial _t^j{v}(x,l)=0, l=\pm T, j=0,1\Bigg \rbrace . \end{aligned}$$

Theorem 2.1

([10]) Under assumption (A.1) of the metric $g=A^{-1}(x),$ there exist constants $C>0$ and $\gamma _*>0$ such that for any $\gamma >\gamma _*,$ there exists $s_0=s(\gamma )$ such that for all $s>s_0>1,$ the following Carleman estimate hold:

$$\begin{aligned} \int _{{Q}}\left[ \sigma (|\nabla _g{v}|_g^2+{v}_t^2)+\sigma ^3{v}^2\right] \textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t\le C\left( \int _{{Q}}|P{v}|^2\textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t+\int _\varSigma \textrm{BT}|_{{\varSigma }}\textrm{d}\varSigma \right) ,\nonumber \\ \end{aligned}$$

(2.19)

whenever ${v}\in {{{\mathcal {H}}}}$ and the right-hand side of (2.19) is finite, with $\sigma =s\gamma \varphi .$ In addition, the boundary terms in $\textrm{BT}|_{\varSigma }$ are given explicitly by

$$\begin{aligned} \textrm{BT}|_{{\varSigma }}= & {} \sigma {z}_t^2\frac{\partial \psi }{\partial \nu _{{\mathcal {A}}}}-2\sigma \frac{\partial {z}}{\partial \nu _{{\mathcal {A}}}}\left( {z}_t\psi _t-\left\langle \nabla _g{z},\nabla _g\psi \right\rangle _g \right) \nonumber \\{} & {} \quad -\sigma \frac{\partial \psi }{\partial \nu _{{\mathcal {A}}}}|\nabla _g{z}|_g^2+\frac{\gamma ^2}{2}\sigma \frac{\partial \psi }{\partial \nu _{{\mathcal {A}}}}{z}^2\left( \psi _t^2-|\nabla _g\psi |_g^2\right) \nonumber \\{} & {} \quad -\gamma \sigma \frac{\partial {z}}{\partial \nu _{{\mathcal {A}}}}{z}\left( \psi _t^2-|\nabla _g\psi |_g^2\right) -\sigma ^3 \frac{\partial \psi }{\partial \nu _{{\mathcal {A}}}}\left( \psi _t^2-|\nabla _g\psi |_g^2\right) {z}^2\nonumber \\{} & {} \quad +\sigma \frac{\partial {z}}{\partial \nu _{{\mathcal {A}}}}{z}(\psi _{tt}-{{\textrm{div}}}A(x)\nabla \psi ), \end{aligned}$$

(2.20)

where ${z}=\textrm{e}^{s\varphi }v.$

We mention that in [10] the boundary terms are given by

$$\begin{aligned} \int _\varSigma \textrm{BT}|_{{\varSigma }}\textrm{d}\varSigma -\frac{\gamma }{2}\int _{{\varSigma }}\sigma {z}^2\frac{\partial |\nabla _g\psi |_g^2}{\partial \nu _{{\mathcal {A}}}}\textrm{d}\varSigma :=\int _\varSigma \textrm{BT}|_{{\varSigma }}\textrm{d}\varSigma -I_1. \end{aligned}$$

(2.21)

Thanks to the inner estimate, the second term $I_1$ on the right-hand side of (2.21) can be absorbed. In fact, $I_1$ arises from the following term (see [10, (40)])

$$\begin{aligned} \gamma \int _{{Q}}\sigma \left\langle \nabla _g{z},\nabla _g|\nabla _g\psi |_g^2 \right\rangle _g{z}\textrm{d}x\textrm{d}t:=I_2. \end{aligned}$$

(2.22)

Since there exists a positive constant $C=C(\psi )$, such that

$$\begin{aligned} I_2\le C\gamma \int _{{Q}}|\nabla _g{z}|_g^2\textrm{d}x\textrm{d}t+\gamma \int _{{Q}}\sigma ^2{z}^2\textrm{d}x\textrm{d}t, \end{aligned}$$

(2.23)

the term $I_1$ is absorbed by the left-hand side of (2.19) when $s>0$ is sufficiently large.

2.2 Observability Inequalities

We consider the observability inequalities for the following system with a non-homogeneous term:

$$\begin{aligned} \left\{ \begin{array}{ll} v_{tt}-\,{{\textrm{div }}}\,A(x)\nabla v=h &{}\quad {\quad \text{ for }\quad }(x,t)\in {Q},\\ \frac{\partial v}{\partial \nu _{{\mathcal {A}}}}=0&{}\quad {\quad \text{ for }\quad }(x,t)\in {\varSigma _1},\\ \frac{\partial v}{\partial \nu _{{\mathcal {A}}}}=f&{}\quad {\quad \text{ for }\quad }(x,t)\in {\varSigma _0},\\ (v(x,0),v_t(x,0))=(v_0,v_1)&{}\quad {\quad \text{ for }\quad }x\in \varOmega . \end{array} \right. \end{aligned}$$

(2.24)

Let ${\mathcal {V}}=L^2(Q)\times L^2(\varSigma _0).$ We have the following.

Theorem 2.2

Assume that the assumption (A.1) holds. Let T satisfy (2.15). Let $(h,f)\in {\mathcal {V}}$ and $(v_0,v_1)\in H^1(\varOmega )\times L^2(\varOmega ).$ Then there exists a constant $C=C(T,C_0)>0$ such that for all $t\in (-T,T),$

$$\begin{aligned} \int _{\varOmega }\left( v^2+|\nabla _gv|^2_g+v_t^2\right) \textrm{d}x\le & {} C \textrm{e}^{-2s(d-\varepsilon )}\int _{Q}h^2\textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t+C\int _Qh^2\textrm{d}x\textrm{d}t\nonumber \\{} & {} \quad + C\textrm{e}^{2sM}\int _{\varSigma _0}[\sigma ^3v^2+\sigma (v_t^2+|\nabla _gv|_g^2)]\textrm{d}\varSigma \nonumber \\ \end{aligned}$$

(2.25)

for $s>0$ large, where $M=\mathop {\sup }\nolimits _{(x,t) \in Q}\varphi (x,t)$.

Proof

Since $(h,f)\in {\mathcal {V}}$ and $(v_0,v_1)\in H^1(\varOmega )\times L^2(\varOmega ),$ the regularity of hyperbolic problems implies that (2.24) admits a unique solution v such that

$$\begin{aligned} v\in H^1(-T,T;L^2(\varOmega ))\cap L^2(-T,T;H^1(\varOmega )). \end{aligned}$$

Let $\textrm{BT}|_\varSigma$ be given by (2.20) and let ${z}=\textrm{e}^{s\varphi }v$. By condition (i) in assumption (A.1) and the boundary condition in (2.24), we have

$$\begin{aligned} \frac{\partial \psi }{\partial \nu _{{{\mathcal {A}}}}}=0,\quad \frac{\partial z}{\partial \nu _{{{\mathcal {A}}}}}=\sigma z\frac{\partial \psi }{\partial \nu _{{{\mathcal {A}}}}}+\textrm{e}^{s\varphi }\frac{\partial v}{\partial \nu _{{{\mathcal {A}}}}}=0 {\quad \text{ for }\quad }(x,t)\in \varSigma _1. \end{aligned}$$

It then follows from (2.20) that

$$\begin{aligned} \int _\varSigma \textrm{BT}|_{\varSigma }\textrm{d}\varSigma \le C\textrm{e}^{2sM}\varGamma ([-T,T],v), \end{aligned}$$

(2.26)

where

$$\begin{aligned} \varGamma ([-T,T],v)=\int _{-T}^{T}\int _{\varGamma _0}\left[ \sigma ^3v^2 +\sigma \left( v_t^2+|\nabla _gv|_g^2\right) \right] \textrm{d}\varGamma \textrm{d}t. \end{aligned}$$

Let

$$\begin{aligned} E(t)=\int _\varOmega \left( v_t^2+|\nabla _gv|_g^2\right) \textrm{d}x+\int _{\varGamma _0}v^2\textrm{d}\varGamma . \end{aligned}$$

By Poincaré’s inequality, we have

$$\begin{aligned} \int _\varOmega \left( v^2+v_t^2+|\nabla _gv|_g^2\right) \textrm{d}x\le CE(t). \end{aligned}$$

(2.27)

For given $\varepsilon >0$ small, we fixed $\delta >0$ small such that (2.17) and (2.18) hold. Taking a cut-off function $\chi (t)\in C_0^2([-T,T])$ satisfying

$$\begin{aligned} \chi (t) = \left\{ \begin{array}{ll} 1,&{}\quad t\in [-T+2\delta ,T-2\delta ], \\ 0, &{}\quad t\in [-T,-T+\delta ]\cup [T-\delta , T]. \end{array} \right. \end{aligned}$$

(2.28)

Then $\chi v\in {{{\mathcal {H}}}}$ and

$$\begin{aligned} P(\chi v)=\chi P(v)+\chi ^{\prime \prime }v+2\chi ^{\prime }v_t=\chi h+\chi ^{\prime \prime }v+2\chi ^{\prime }v_t. \end{aligned}$$

(2.29)

Applying the Carleman estimate (2.19) to (2.29), for $T\ge 3\delta ,$ we obtain, by (2.26), the following:

$$\begin{aligned}{} & {} \textrm{e}^{2s(d-\varepsilon )}\int _{-\delta }^{\delta }\int _\varOmega (v^2+v_t^2+|\nabla _gv|_g)^2\textrm{d}x\textrm{d}t\nonumber \\{} & {} \quad \le \int _{Q}\left( |\nabla _g(\chi v)|_g^2+|(\chi v)_t|^2+|\chi v|^2 \right) \textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t\nonumber \\{} & {} \quad \le C \int _{Q}(|\chi h|^2+|\chi ^{\prime \prime }v|^2+|\chi ^{\prime }v_t|^2) \textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t+C\textrm{e}^{2sM}\varGamma ([-T,T],\chi v)\nonumber \\{} & {} \quad \le C\int _Q|h|^2\textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t+C\left( ||v||^2_{L^2(Q)}+||v_t||^2_{L^2(Q)}\right) \textrm{e}^{2s(d-2\varepsilon )}\nonumber \\{} & {} \qquad +C\textrm{e}^{2sM}\varGamma ([-T+\delta ,T-\delta ],v), \end{aligned}$$

(2.30)

where we have used the inequality $\varphi \le d-2\varepsilon$ only in the case where $\chi ^{\prime }\ne 0.$ By the mean value theorem, there exists a $t_1\in (-\delta ,\delta )$ such that

$$\begin{aligned} \int _{\varOmega }\left( |\nabla _gv|_g^2+v_t^2+v^2\right) \textrm{d}x\Big |_{t=t_1}\le & {} C\textrm{e}^{-2s(d-\varepsilon )} \int _Q|h|^2\textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t\nonumber \\{} & {} \quad + C\textrm{e}^{2s(M-d+\varepsilon )}\varGamma ([-T+\delta ,T-\delta ],f)\nonumber \\{} & {} \quad +C\textrm{e}^{-2s\varepsilon }\left( ||v||^2_{L^2(Q)}+||v_t||^2_{L^2(Q)}\right) .\nonumber \\ \end{aligned}$$

(2.31)

It then follows from (2.27) and the standard energy integration that

$$\begin{aligned} \int _{\varOmega }\left( v^2+v_t^2+|\nabla _gv|_g^2\right) \textrm{d}x\le & {} C\int _{\varOmega } \left( v^2+v_t^2+|\nabla _gv|_g^2\right) \textrm{d}x\Big |_{t=t_1}\nonumber \\{} & {} \quad +C\int _{\varSigma _0}\left( v_t^2 +f^2+v^2\right) \textrm{d}\varSigma +C\int _Q|h|^2\textrm{d}x\textrm{d}t.\nonumber \\ \end{aligned}$$

(2.32)

Then

$$\begin{aligned} \int _Q\left( v^2+v_t^2+|\nabla _gv|_g^2\right) \textrm{d}x\textrm{d}t\le & {} CT\int _{\varOmega }\left( v^2+v_t^2+|\nabla _gv|_g^2\right) \textrm{d}x\Big |_{t=t_1}\nonumber \\{} & {} \quad +CT\int _{\varSigma _0}\left( v_t^2+f^2+v^2\right) \textrm{d}\varSigma \nonumber \\{} & {} \quad +CT\int _Q|h|^2\textrm{d}x\textrm{d}t. \end{aligned}$$

(2.33)

Taking s large enough. By (2.31) and (2.33), the term $C\textrm{e}^{-2s\varepsilon }\left( ||v||^2_{L^2(Q)}+||v_t||^2_{L^2(Q)}\right)$ on the right-hand side of (2.31) is absorbed. Thus, it follows (2.32) that

$$\begin{aligned}{} & {} \int _\varOmega (|\nabla _gv|_g^2+v_t^2+v^2)\textrm{d}x\nonumber \\{} & {} \le C\textrm{e}^{-2s(d-\varepsilon )} \int _Q|h|^2\textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t\nonumber \\{} & {} \quad +C\int _Qh^2\textrm{d}x\textrm{d}t+ C\textrm{e}^{2s(M-d+\varepsilon )}\varGamma ([-T+\delta ,T-\delta ],v), \end{aligned}$$

(2.34)

and hence (2.25) follows. $\square$

The following lemma is quoted from [19], from which the tangential derivative $\nabla _{\varGamma _g }v$ on $\varSigma _0$ can be removed from the right-hand side of (2.25) in the case where $h=0.$

Lemma 2.1

([19]) Let v solve problem (2.24) with $h=0.$ Then, for given small $\delta :$ $0<\delta <T,$ there exists a positive constant $C=C(T,\delta )$ such that

$$\begin{aligned} \int _{-T+\delta }^{T-\delta }\int _{\varGamma _0}\left| \nabla _{\varGamma _g}v\right| ^2 \textrm{d}\varGamma \textrm{d}t\le C\int _{\varSigma _0}\left( v_t^2+\left| \frac{\partial v}{\partial \nu _{{\mathcal {A}}}} \right| ^2 \right) \textrm{d}\varSigma +L(v), \end{aligned}$$

(2.35)

where L(v) denotes the lower-order terms of v with respect to the norm of $C(-T,T;H^1(\varOmega )).$

Using (2.35) in (2.34) and by a compactness–uniqueness argument, we have the following.

Corollary 2.1

Let v solve problem (2.24) with $h=0.$ Then for $s>0$ large,

$$\begin{aligned} \int _{\varOmega }\left( v^2+|\nabla _gv|^2_g+v_t^2\right) \textrm{d}x\le C\textrm{e}^{2sM}\int _{\varSigma _0}\left( v^2+v_t^2+f^2\right) \textrm{d}\varSigma . \end{aligned}$$

(2.36)

3 Proofs of the Main Theorems

A similar argument as in the proof of [9, Lemma 3.1] yields the following lemma.

Lemma 3.1

Let assumption (A.2) hold. Then there is a $C>0$ such that

$$\begin{aligned} \int _{\varOmega }(|B(x)|^2+|\nabla B(x)|^2)\textrm{e}^{2s\varphi (x,0)}\textrm{d}x\le C\int _{\varOmega }(|J|^2+|\nabla J|^2)\textrm{e}^{2s\varphi (x,0)}\textrm{d}x \end{aligned}$$

(3.1)

for $s>0$ large, where $B(x)=(b_{ij}(x))_{1\le i,j\le 3}$ and

$$\begin{aligned} J(x)=({{\textrm{div}}}B(x)\nabla a_1(x),\ldots ,{{\textrm{div}}}B(x)\nabla a_9(x))^\textrm{T}. \end{aligned}$$

Let (W, U) solve problem (1.2). Set

$$\begin{aligned} {\overline{W}}=W_t.\quad \end{aligned}$$

By (1.2), $({{\overline{W}}}, U)$ satisfies problem

$$\begin{aligned} \left\{ \begin{array}{ll} {\overline{W}}_{tt}-{\,{{\textrm{div }}}\,}A_1(x)\nabla {{\overline{W}}}={{\textrm{div}}}B\nabla R_t &{}\quad {\quad \text{ in }\quad }\varOmega \times (0,T),\\ \frac{\partial {{\overline{W}}}}{\partial \nu _{{\mathcal {A}}_1}}=0&{}\quad \quad \text{ on }\quad \varGamma _1\times (0,T),\\ \frac{\partial {{\overline{W}}}}{\partial \nu _{{\mathcal {A}}_1}}=U_{tt}&{}\quad \quad \text{ on }\quad \varGamma _0\times (0,T),\\ U_{tt}+{\mathcal {A}}_0^2U=-{{\overline{W}}}&{}\quad \quad \text{ on }\quad \varGamma _0\times (0,T),\\ U=\frac{\partial U}{\partial n_0}=0 &{}\quad \quad \text{ on }\quad \partial \varGamma _0 \times (0,T) ,\\ {{\overline{W}}}(x,0)=0, \ {{\overline{W}}}_t(x,0)=J(x) &{}\quad {\quad \text{ in }\quad }\ \varOmega ,\\ U(x,0)=U_t(x,0)=0 &{}\quad \quad \text{ on }\quad \varGamma _0. \end{array} \right. \end{aligned}$$

(3.2)

Proof of Theorem 1.1

Let $(\overline{W},U)$ solve problem (3.2). Let $U_{tt}(x,t)=0$ on $\varSigma _0.$ We proceed to prove that

$$\begin{aligned} B(x)=0{\quad \text{ for }\quad }x\in \varOmega \end{aligned}$$

holds as follows.

The assumptions $U_{tt}(t,x)=0$ for $(t,x)\in \varSigma _0$ and $U(x,0)=U_t(x,0)=0$ for $x\in \varGamma _0$ imply that

$$\begin{aligned} U(t,x)=0{\quad \text{ for }\quad }(t,x)\in \varSigma _0. \end{aligned}$$

Let

$$\begin{aligned} {{\widetilde{W}}}={\overline{W}}_t,\quad \widetilde{\widetilde{W}}=\overline{W}_{tt}{\quad \text{ for }\quad }(t,x)\in Q. \end{aligned}$$

It is easy to check from (3.2) that $\widetilde{W}$ satisfies the following:

$$\begin{aligned} \left\{ \begin{array}{ll} {{\widetilde{W}}}_{tt}-{\,{{\textrm{div }}}\,}A_1(x)\nabla {{\widetilde{W}}}=\,{{\textrm{div }}}\,B\nabla R_{tt} &{} {\quad \text{ in }\quad }\varOmega \times (0,T),\\ \frac{\partial {{\widetilde{W}}}}{\partial \nu _{{\mathcal {A}}_1}}=0&{} \quad \text{ on }\quad \varGamma _1\times (0,T),\\ \frac{\partial {{\widetilde{W}}}}{\partial \nu _{{\mathcal {A}}_1}}=\widetilde{W}=0 &{} \quad \text{ on }\quad \varGamma _0\times (0,T),\\ {{\widetilde{W}}}(x,0)=J(x), \ {{\widetilde{W}}}_t(x,0)=0 &{} {\quad \text{ in }\quad }\varOmega , \end{array} \right. \end{aligned}$$

(3.3)

and $\widetilde{\widetilde{W}}$ solves the following:

$$\begin{aligned} \left\{ \begin{array}{ll} \widetilde{\widetilde{W}}_{tt}-{\,{{\textrm{div }}}\,}A_1(x)\nabla \widetilde{\widetilde{W}}=\,{{\textrm{div }}}\,B\nabla R_{ttt} &{} {\quad \text{ in }\quad }\varOmega \times (0,T),\\ \frac{\partial \widetilde{\widetilde{W}}}{\partial \nu _{{\mathcal {A}}_1}}=0&{} \quad \text{ on }\quad \varGamma _1\times (0,T),\\ \frac{\partial \widetilde{\widetilde{W}}}{\partial \nu _{{\mathcal {A}}_1}}=\widetilde{\widetilde{W}}=0 &{} \quad \text{ on }\quad \varGamma _0\times (0,T),\\ \widetilde{\widetilde{W}}(x,0)=0, \,\widetilde{\widetilde{W}}_t(x,0)={{\hat{F}}}(x) &{} {\quad \text{ in }\quad }\varOmega , \end{array} \right. \end{aligned}$$

(3.4)

where

$$\begin{aligned} {{\hat{F}}}(x)=\,{{\textrm{div }}}\,A_1(x)\nabla J(x)+{{\textrm{div}}}B(x)\nabla R_{tt}(x,0)\in L^2(\varOmega ), \end{aligned}$$

since $A_1(x)$ and $A_2(x)$ are in ${\mathcal {U}}(C_0)$.

Let the cut-off function $\chi (t)$ given by (2.28). We apply the Carleman estimate in Theorem 2.1 with

$$\begin{aligned} P(\chi \widetilde{W})=\partial _{t}^2(\chi \widetilde{W})-\,{{\textrm{div }}}\,A_1(x)\nabla (\chi \widetilde{W})=\chi ''{\widetilde{W}}+2\chi '{\widetilde{W}}_t+\chi \,{{\textrm{div }}}\,B\nabla R_{tt} \end{aligned}$$

to (3.3), to obtain

$$\begin{aligned}{} & {} \int _{Q}\left[ \sigma \left( |(\chi {{\widetilde{W}}})_t|^2+|\chi \nabla _g{{\widetilde{W}}}|_g^2\right) +\sigma ^3|\chi {{\widetilde{W}}}|^2\right] \textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t\nonumber \\{} & {} \quad \le C\int _{Q}|\,{{\textrm{div }}}\,B\nabla R_{tt} |^2\textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t+C\textrm{e}^{2s(d-2\varepsilon )}\int _{Q}\left( |{{\widetilde{W}}}_t|^2+|{{\widetilde{W}}}|^2\right) \textrm{d}x\textrm{d}t,\nonumber \\ \end{aligned}$$

(3.5)

where $\sigma =s\,{{\mathrm{\gamma }}}\,\varphi .$ Similarly, applying Theorem 2.1 to (3.3) yields

$$\begin{aligned}{} & {} \int _{Q}\left[ \sigma \left( |(\chi \widetilde{\widetilde{W}})_t|^2+|\chi \nabla _g\widetilde{\widetilde{W}}|_g^2\right) +\sigma ^3|\chi \widetilde{\widetilde{W}}|^2\right] \textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t\nonumber \\{} & {} \quad \le C\int _{Q}|\,{{\textrm{div }}}\,B\nabla R_{ttt} |^2\textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t+C\textrm{e}^{2s(d-2\varepsilon )}\int _{Q}( |\widetilde{\widetilde{W}}_t|^2+|\widetilde{\widetilde{W}}|^2) \textrm{d}x\textrm{d}t.\nonumber \\ \end{aligned}$$

(3.6)

Next, since $\widetilde{W}(x,0)=J(x),$ by (3.5) we have

$$\begin{aligned} \int _\varOmega |J(x)|^2\textrm{e}^{2s\varphi (x,0)}\textrm{d}x= & {} \int _{-T}^0\frac{\partial }{\partial t}\int _\varOmega |\chi (t)\widetilde{W}(x,t)|^2\textrm{e}^{2s\varphi (x,t)}\textrm{d}x\textrm{d}t\nonumber \\\le & {} C\int _Q(|\chi '||\widetilde{W}|^2+\sigma |\chi \widetilde{W}|^2+|(\chi \widetilde{W})_t|^2)\textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t\nonumber \\\le & {} C\int _{Q}|\,{{\textrm{div }}}\,B\nabla R_{tt} |^2\textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t\nonumber \\{} & {} +C\textrm{e}^{2s(d-2\varepsilon )}\int _{Q}( |{{\widetilde{W}}}_t|^2+|{{\widetilde{W}}}|^2) \textrm{d}x\textrm{d}t. \end{aligned}$$

(3.7)

Moreover, since

$$\begin{aligned} \varsigma |\nabla J(x)|\le |\nabla _gJ(x)|\le C|\nabla J(x)|{\quad \text{ for }\quad }x\in \varOmega \end{aligned}$$

for some $\varsigma >0$ small, it follows from (3.5) and (3.6) that

$$\begin{aligned}{} & {} \int _\varOmega |\nabla J(x)|^2 \textrm{e}^{2s\varphi (x,0)}\textrm{d}x\nonumber \\\le & {} C\int _\varOmega |\nabla _gJ(x)|_g^2\textrm{e}^{2s\varphi (x,0)}\textrm{d}x\nonumber \\= & {} C\int _{-T}^0\frac{\partial }{\partial t}\int _\varOmega |\chi \nabla _g\widetilde{W}|_g^2\textrm{e}^{2s\varphi (x,t)}\textrm{d}x\textrm{d}t\nonumber \\\le & {} C\int _Q\left( |\chi '||\nabla _g\widetilde{W}|_g^2+\sigma |\chi \nabla _g\widetilde{W}|_g^2+|\chi \nabla _g\widetilde{\widetilde{W}}|_g^2\right) \textrm{e}^{2s\varphi (x,t)}\textrm{d}x\textrm{d}t\nonumber \\\le & {} C\int _{Q}(|\,{{\textrm{div }}}\,B\nabla R_{tt} |^2+|\,{{\textrm{div }}}\,B\nabla R_{ttt} |^2)\textrm{e}^{2s\varphi }\textrm{d}x\textrm{d}t\nonumber \\{} & {} \quad +C\textrm{e}^{2s(d-2\varepsilon )}\int _{Q}(|\nabla _g\widetilde{W}|_g^2+|{{\widetilde{W}}}_t|^2+|{{\widetilde{W}}}|^2+|\widetilde{\widetilde{W}}_t|^2+|\widetilde{\widetilde{W}}|^2) \textrm{d}x\textrm{d}t.\nonumber \\ \end{aligned}$$

(3.8)

On the other hand, assumption (1.8) implies

$$\begin{aligned} |\,{{\textrm{div }}}\,B\nabla R_{tt} |^2+|\,{{\textrm{div }}}\,B\nabla R_{ttt} |^2\le C(|J(x)|^2+|\nabla J(x)|^2){\quad \text{ for }\quad }(t,x)\in Q. \end{aligned}$$

From (3.7) and (3.8), we obtain

$$\begin{aligned}{} & {} \int _\varOmega (|J(x)|^2+|\nabla J(x)|^2)\textrm{e}^{2s\varphi (x,0)}\textrm{d}x\nonumber \\{} & {} \quad \le C\int _\varOmega (|J(x)|^2+|\nabla J(x)|^2)\textrm{e}^{2s\varphi (x,0)}|\int _{-T}^\textrm{T}\textrm{e}^{2s[\varphi (x,t)-\varphi (x,0)]}\textrm{d}t|\textrm{d}x\nonumber \\{} & {} \qquad +C\textrm{e}^{2s(d-2\varepsilon )}\int _{Q}(|\nabla _g\widetilde{W}|_g^2+|{{\widetilde{W}}}_t|^2+|{{\widetilde{W}}}|^2+|\widetilde{\widetilde{W}}_t|^2+|\widetilde{\widetilde{W}}|^2) \textrm{d}x\textrm{d}t.\nonumber \\ \end{aligned}$$

(3.9)

We assume that there is a small number $c>0$ such that

$$\begin{aligned} \textrm{e}^{-c\,{{\mathrm{\gamma }}}\,t^2}-1\le -c\varsigma \,{{\mathrm{\gamma }}}\,t^2{\quad \text{ for }\quad }t\in (-T,T) \end{aligned}$$

for some $\varsigma >0$ small. It is easy to check that

$$\begin{aligned} \sup _{x\in \varOmega }\left| \int _{-T}^{T}\textrm{e}^{2s[\varphi (x,t)-\varphi (x,0)]}\textrm{d}t\right| \rightarrow 0{\quad \text{ at }\quad }s\rightarrow \infty . \end{aligned}$$

Thus, the first term on the right-hand side of (3.9) can be absorbed by the left-hand side of (3.9). By (3.1) and (3.9), we obtain

$$\begin{aligned} \int _{\varOmega }(|B(x)|^2+|\nabla B(x)|^2)\textrm{e}^{2sd}\textrm{d}x\le & {} \int _{\varOmega }(|B(x)|^2+|\nabla B(x)|^2)\textrm{e}^{2s\varphi (x,0)}\textrm{d}x\nonumber \\\le & {} C\textrm{e}^{2s(d-2\varepsilon )}\int _{Q}(|\nabla _g\widetilde{W}|_g^2+|{{\widetilde{W}}}_t|^2+|{{\widetilde{W}}}|^2\nonumber \\{} & {} +|\widetilde{\widetilde{W}}_t|^2+|\widetilde{\widetilde{W}}|^2) \textrm{d}x\textrm{d}t \end{aligned}$$

(3.10)

for $s>0$ large. From (3.10), the proof of Theorem 1.1 is complete by taking $s\rightarrow \infty .$ $\square$

Proof of Theorem 1.2

Let $(\overline{W},U)$ solve problem (3.2). Set $\widetilde{W}=\overline{W}_t.$ By (3.2) $(\widetilde{W},U)$ solves problem

$$\begin{aligned} \left\{ \begin{array}{ll} {{\widetilde{W}}}_{tt}-{\,{{\textrm{div }}}\,}A_1\nabla {{\widetilde{W}}}={{\textrm{div}}}B\nabla R_{tt} &{} {\quad \text{ in }\quad }\varOmega \times (0,T),\\ \frac{\partial {{\widetilde{W}}}}{\partial \nu _{{\mathcal {A}}_1}}=0 &{} \quad \text{ on }\quad \varGamma _1\times (0,T),\\ \frac{\partial {{\widetilde{W}}}}{\partial \nu _{{\mathcal {A}}_1}}=\partial _t^3 U &{} \quad \text{ on }\quad \varGamma _0\times (0,T),\\ \partial _t^3 U+{\mathcal {A}}_0^2U_t=-{{\widetilde{W}}} &{} \quad \text{ on }\quad \varGamma _0\times (0,T),\\ U=\frac{\partial U}{\partial n_0}=0 &{} \quad \text{ on }\quad \partial \varGamma _0 \times (0,T) ,\\ {{\widetilde{W}}}(x,0)=J, \ {{\widetilde{W}}}_t(x,0)=0 &{} {\quad \text{ in }\quad }\varOmega ,\\ U(x,0)=U_t(x,0)=U_{tt}(x,0)=0 &{} \quad \text{ on }\quad \varGamma _0. \end{array} \right. \end{aligned}$$

(3.11)

Noting that the assumption

$$\begin{aligned} A_1(x)=A_2(x)=A_0(x){\quad \text{ for }\quad }x\in \varGamma _0, \end{aligned}$$

we have

$$\begin{aligned} J(x)=0{\quad \text{ for }\quad }x\in \varGamma _0. \end{aligned}$$

Then

$$\begin{aligned} \partial _t^3U(x,0)=-J(x)-{{{\mathcal {A}}}}_0^2U_t(x,0)=0{\quad \text{ for }\quad }x\in \varGamma _0. \end{aligned}$$

Thus

$$\begin{aligned} \Vert \partial _t^3U\Vert _{L^2(\varSigma _0)}\le C \Vert \partial _t^4U\Vert _{L^2(\varSigma _0)},\quad \Vert {{{\mathcal {A}}}}_0^2U_t\Vert _{L^2(\varSigma _0)}\le C \Vert {{{\mathcal {A}}}}_0^2U_{tt}\Vert _{L^2(\varSigma _0)}. \end{aligned}$$

For given U by (3.11), we define $\varPhi$ as the unique solution to the following problem:

$$\begin{aligned} \left\{ \begin{array}{ll} \varPhi _{tt}-{\,{{\textrm{div }}}\,}A_1\nabla \varPhi =0 &{} \quad {{\textrm{in}}} \ \varOmega \times (0,T),\\ \frac{\partial \varPhi }{\partial \nu _{{\mathcal {A}}_1}}=0 &{}{\quad \text{ in }\quad }\varGamma _1\times (0,T),\\ \frac{\partial \varPhi }{\partial \nu _{{\mathcal {A}}_1}}=\partial _t^3 U &{}\quad \text{ on }\quad \varGamma _0\times (0,T),\\ \varPhi (x,0)=J(x),\, \varPhi _t(x,0)=0 &{}{\quad \text{ in }\quad }\varOmega . \end{array} \right. \end{aligned}$$

(3.12)

We apply Theorem 2.2 to problem (3.12) with $v=\varPhi ,$ and recall B in (3.1), to have

$$\begin{aligned}{} & {} \Vert B\Vert ^2_{H^1(\varOmega )}\nonumber \\\le & {} C\left( ||\partial _t^3U||^2_{L^2(\varSigma _0)}+||\varPhi ||^2_{L^2(\varSigma _0)}+||\varPhi _t||^2_{L^2(\varSigma _0)}\right) \nonumber \\\le & {} C\left( ||\partial _t^3U||^2_{L^2(\varSigma _0)}+||Y||^2_{L^2(\varSigma _0)}+||Y_t||^2_{L^2(\varSigma _0)}+\Vert \widetilde{W}\Vert ^2_{L^2(\varSigma _0)}+\Vert \widetilde{W}_t\Vert ^2_{L^2(\varSigma _0)}\right) \nonumber \\\le & {} C\left( ||\partial _t^4U||^2_{L^2(\varSigma _0)}+\Vert {{{\mathcal {A}}}}_0^2U_{tt}\Vert ^2_{L^2(\varSigma _0)}+||Y||^2_{L^2(\varSigma _0)}+||Y_t||^2_{L^2(\varSigma _0)}\right) ,\nonumber \\ \end{aligned}$$

(3.13)

where

$$\begin{aligned} Y=\widetilde{W}-\varPhi {\quad \text{ for }\quad }(t,x)\in Q, \end{aligned}$$

that solves problem

$$\begin{aligned} \left\{ \begin{array}{ll} Y_{tt}-{\,{{\textrm{div }}}\,}A_1\nabla Y={{\textrm{div}}}B\nabla R_{tt} &{} {\quad \text{ in }\quad }\varOmega \times (0,T),\\ \frac{\partial Y}{\partial \nu _{{\mathcal {A}}_1}}=0&{} \quad \text{ on }\quad \varGamma \times (0,T),\\ Y(x,0)=0, \quad Y_t(x,0)=0 &{} {\quad \text{ in }\quad }\varOmega , \end{array} \right. \end{aligned}$$

(3.14)

by (3.11) and (3.12).

Next, we remove the term $||Y||^2_{L^2(\varSigma _0)}+||Y_t||^2_{L^2(\varSigma _0)}$ from the right-hand side of (3.13) by a compactness–uniqueness argument below as in [23] (see also [24]).

For simplicity, denote

$$\begin{aligned} ||U||^2_{S}=||\partial _t^4U||^2_{L^2(\varSigma _0)}+||{\mathcal {A}}_0^2U_{tt}||^2_{L^2(\varSigma _0)}. \end{aligned}$$

We define a map ${{{\mathcal {K}}}}:$ $H^1(\varOmega )\rightarrow L^2(\varSigma _0)\times L^2(\varSigma _0)$ by

$$\begin{aligned} {{{\mathcal {K}}}}B=(Y,Y_t), \end{aligned}$$

where Y solves problem (3.14) for given B(x).

Since the initial data $(a_k,0,v_0,0)\in {\mathcal {F}},$ where ${\mathcal {F}}$ is given by (2.11), the semigroup theory gives that

$$\begin{aligned} (R,R_t,V_2,V_{2t})\in C((-T,T);D(\varLambda ^6)). \end{aligned}$$

Therefore, we deduce that

$$\begin{aligned} \partial ^6_tR\in C((-T,T);H^1(\varOmega )),\quad \partial _t^7R\in C((-T,T);L^2(\varOmega )). \end{aligned}$$

Since ${\mathcal {A}}(\partial _t^5R)=(\partial _t^5R)_{tt}\in C((-T,T);L^2(\varOmega )),$ elliptic theory yields

$$\begin{aligned} \partial _t^5R\in C((-T,T);H^2(\varOmega )). \end{aligned}$$

By the Sobolev embedding theorems for dimension $n=3$, we obtain

$$\begin{aligned}{} & {} \partial _t^2R\in C((-T,T);H^5(\varOmega ))\rightarrow L^{\infty }((-T,T);W^{2,\infty }(\varOmega )),\ {{\textrm{continuously}}};\\{} & {} \partial _t^3R\in C((-T,T);H^4(\varOmega ))\rightarrow L^{\infty }((-T,T);W^{2,\infty }(\varOmega )),\ {{\textrm{continuously}}}. \end{aligned}$$

It is easy to check from (3.14) that

$$\begin{aligned} ||R_{tt}||_{L^{\infty }(-T,T;W^{2,\infty }(\varOmega ))} +||R_{ttt}||_{L^{\infty }(-T,T;W^{2,\infty }(\varOmega ))}<+\infty , \end{aligned}$$

which implies that

$$\begin{aligned} \,{{\textrm{div }}}\,B(x)\nabla R_{tt}\in L^2(Q),\quad \partial _t\,{{\textrm{div }}}\,B(x)\nabla R_{tt}\in L^2(Q). \end{aligned}$$

As a consequence, operator ${{{\mathcal {K}}}}:$ $H^1(\varOmega )\rightarrow L^2(\varSigma _0)\times L^2(\varSigma _0)$ is compact.

We proceed to complete the proof by contradiction. By assumption (1.7), suppose that there exists a sequence $\left\{ B_n \right\} _{n\ge 1} \in H^1(\varOmega )$ such that

$$\begin{aligned} ||B_n||_{H^1(\varOmega )}=1,\quad n\ge 1, \end{aligned}$$

(3.15)

and

$$\begin{aligned} ||Y_n||^2_{L^2(\varSigma _0)}+||Y_{nt}||^2_{L^2(\varSigma _0)}\ge n||U_n||^2_S, \end{aligned}$$

(3.16)

where $Y_n$ and $U_n$ are given by (3.14) and (3.11), respectively, with $B=B_n.$ Then we have

$$\begin{aligned} \mathop {\lim }\limits _{n \rightarrow +\infty }||U_n||_S=0. \end{aligned}$$

(3.17)

By (3.15), there exists a subsequence, still denoted by $\left\{ B_n \right\} _{n\ge 1}$, such that

$$\begin{aligned} B_n\rightharpoonup B_0\in H^1(\varOmega )\,\text{ weakly } \text{ in },\, H^1(\varOmega ), \end{aligned}$$

(3.18)

for some $B_0\in H^1(\varOmega ).$ Let $(\widetilde{W}_n,U_n)$ and $(\widetilde{W}_0,U_0)$ be given by (3.11) with $B=B_n$ and $B=B_0,$ respectively. It follows from (3.13) that

$$\begin{aligned} B_n\rightarrow B_0\in H^1(\varOmega )\,\text{ strongly } \text{ in } \, H^1(\varOmega ), \end{aligned}$$

and $\Vert B_0\Vert _{H^1(\varOmega )}=1.$

By the trace theorem and an a priori estimate of (3.11), we obtain

$$\begin{aligned} \Vert \widetilde{W}_n\Vert _{L^2(\varSigma _0)}\le C\Vert \widetilde{W}_n\Vert _{H^{1/2}(Q)}\le C\Vert \widetilde{W}_n\Vert _{H^1(Q)}\le C\left( ||U_n||_S+||B_n||_{H^1(\varOmega )}\right) , \end{aligned}$$

yielding

$$\begin{aligned} {{\widetilde{W}}}_n\rightarrow {{\widetilde{W}}}_0\,\text{ strongly } \text{ in }\, L^2(\varSigma _0). \end{aligned}$$

(3.19)

Thus

$$\begin{aligned} \Vert \widetilde{W}_0\Vert _{L^2(\varSigma _0)}=\mathop {\lim }\limits _{n \rightarrow \infty } ||{{\widetilde{W}}}_n||_{L^2(\varSigma _0)}\le C\mathop {\lim }\limits _{n \rightarrow \infty }||U_n||_S=0, \end{aligned}$$

that is,

$$\begin{aligned} \partial _t^3U_0+{\mathcal {A}}_0^2U_0=0. \end{aligned}$$

By (3.11), $U_0=\varPsi _t$ solves the following:

$$\begin{aligned} \left\{ \begin{array}{ll} \varPsi _{tt}+{\mathcal {A}}_0^2\varPsi =0 &{} {\quad \text{ in }\quad }\varSigma _0,\\ \varPsi =\frac{\partial \varPsi }{\partial n_0}=0&{} \quad \text{ on }\quad \partial \varGamma _0,\\ \varPsi (x,0)=\varPsi _t(x,0)=0&{} {\quad \text{ in }\quad }\varGamma _0. \end{array} \right. \end{aligned}$$

(3.20)

The uniqueness of problem (3.20) implies

$$\begin{aligned} U_0=0\quad \text{ on }\quad \varSigma _0. \end{aligned}$$

By Theorem 1.1, $B_0=0,$ which contradicts with the fact that $||B_0||_{H^1(\varOmega )}=1.$ $\square$

4 Concluding Remarks

The main prominent feature of the structural acoustic system (1.1) lies in the presence of a variable coefficient matrix A(x),  which arises naturally from the non-homogeneous material properties. We may further study the inverse problems for the structural model with a curved wall whose middle surface is a part of a surface in $\textrm{IR}^3$. For the modeling of the structural acoustic systems with variable coefficients and curved walls, we refer to Appendix in [33]. The above two characters not only make the structural acoustic system much more realistic, but also gain additional complexities to the mathematical analysis.

We mention that all the results obtained in this paper are also valid for the case where the dimension $n=2.$ That is, the plate $\varGamma _0$ reduces to the beam. It is also pointed out that the observability inequality (2.36) obtained by the Carleman estimate can also be proved by the well-known multiplier technique only. See for example [34, Chap. 2].

Assumption (A.2) means that we need to repeat observations 9 times for the determination of 6 unknown coefficients $(a_{ij}(x))_{1\le i,j\le 3}$. An interesting question is: Can we suitably choose 6 or less groups of inputs (observations) for determining $(a_{ij}(x))_{1\le i,j\le 3}$? However, we do not know how to achieve this. Anyways, this needs further considerations, and some estimates (e.g., Lemma 3.1) should be refined.