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Asymptotic stability for a strongly coupled Klein-Gordon system in an inhomogeneous medium with locally distributed damping. (English) Zbl 1429.35156

The authors study the exponential stability of a following semilinear wave equation system: \[ \begin{cases} \rho(x) u_{tt} - \operatorname{div}(K(x) \nabla u) + v^2u- \operatorname{div}(a(x) \nabla u_t)=0 &~\text{ in }~ \Omega \times (0,T), \\ \rho(x) v_{tt} - \operatorname{div}(K(x) \nabla v) + u^2v - \operatorname{div}(b(x) \nabla v_t)=0 &~\text{ in }~ \Omega\times (0,T), \\ u = v = 0 &~\text{ on }~\partial \Omega \times (0,T),\\ u(x,0)=u_0(x), ~ u_t(x,0)=u_1(x) &~\text{ in }~ \Omega,\\ v(x,0)=v_0(x), ~ v_t(x,0)=v_1(x) &~\text{ in }~ \Omega,\tag{1} \end{cases} \] where \(\Omega \subset \mathbb{R}^{d}\), \(d \leqslant 3\), is a bounded domain with a sufficiently smooth boundary, \(\rho:\Omega \rightarrow \mathbb{R}_+\), \(k_{ij}:\Omega \rightarrow \mathbb{R}\), \(1\leqslant i,j\leqslant d\) are \(C^\infty(\Omega)\) functions such that \[ \alpha_0\leqslant \rho(x)\leqslant \beta_0,\quad k_{ij}(x) = k_{ji}(x),\quad \alpha|\xi|^2 \leqslant \xi^{\top}\cdot K(x) \cdot \xi \leqslant \beta |\xi|^2, \] \(\alpha_0\), \(\beta_0\), \(\alpha\), \(\beta\) are positive constants and \(K(x)=(k_{ij})_{i,j}\) is a symmetric positive-definite matrix and the function \(\gamma\in W^{1,\infty}(\Omega)\) obeys \[ 0\leqslant \gamma(x) \leqslant a(x),\qquad 0\leqslant \gamma(x)\leqslant b(x)\qquad\text{a.e. in}\ \Omega. \]
The symbol \(\omega\) denotes the intersection of \(\Omega\) with a neighborhood of \(\partial \Omega\) in \(\mathbb{R}^d\); the boundary \(\partial \omega\) is supposed to be smooth.
A series of assumptions is made for the considered problem: The non-negative functions \(a\) and \(b\), responsible for the localized dissipative effect, satisfies the following conditions: \[ a, ~ b \in C^0(\overline{\Omega}) \text{ with } a(x)\geqslant a_0> 0 \text{ in } \omega\subset \Omega \text{ and } b(x)\geqslant b_0> 0 \text{ in } \omega\subset \Omega. \]
\(\omega\) geometrically controls \(\Omega\), i.e there exists \(T_0 >0\), such that every geodesic of the metric \(G(x)\), where \(G(x)=\left(\frac{K(x)}{\rho(x)}\right)^{-1}\) travelling with speed \(1\) and issued at \(t = 0\), intercepts \(\omega\) in a time \(t < T_0\).
For every \(T > 0\), the only solution \(u\), \(v \in C(]0,T[; L^2(\Omega)) \cap C(]0,T[,H^{-1}(\Omega))\) to the system \[ \begin{cases} \rho(x) u_{tt} - \operatorname{div}(K(x) \nabla u) + V_1(x,t)u=V_3(x,t)v &~\text{ in }~ \Omega \times (0,T), \\ \rho(x) v_{tt} - \operatorname{div}[K(x) \nabla v] + V_2(x,t)v=V_3(x,t)u &~\text{ in }~ \Omega\times (0,T), \\ u=v=0 &~\text{ on }~ \omega, \end{cases} \] where \(V_1(x,t)\), \(V_2(x,t)\) and \(V_3(x,t)\) are elements of \(L^\infty( ]0,T[,L^{\frac{d+1}{2}}(\Omega))\), is the trivial one \(u=v= 0\).
Extra made assumptions are
(i)
\(\forall x\in \partial\Omega\), \(a(x), b(x)>0\),
(ii)
For all geodesic \(t\in I\mapsto x(t)\in\Omega\) of the metric \(G=(K/\rho)^{-1}\), with \(0\in I\), there exists \(t\geqslant 0\) such that \((a(x(t)),b(x(t)))>0\).

The main results establish existence and uniqueness for weak solutions to problem (1) and, in addition, that those solutions decay exponentially and uniformly to zero. Namely, assume that the initial data satisfies \[ (u_0,v_0,u_1,v_1)\in \mathcal{H}:=H_{0}^{1}(\Omega) \times H_{0}^{1}(\Omega) \times L^{2}(\Omega) \times L^{2}(\Omega). \] Then the considered problem possesses a unique weak solution in \(C([0,T];\mathcal{H})\). If \((u_0,v_0,u_1,v_1)\in \big(H^2(\Omega)\cap H_{0}^{1}(\Omega)\big)^2 \times \big(H_{0}^{1}(\Omega)\big)^2\), this weak solution is regular. Let \begin{align*} E_{u,v}(t)= & \frac{1}{2}\int_\Omega \rho(x)| u_t(x,t)|^2+\rho(x)|v_t(x,t)|^2 + \nabla u(x,t)^{\top} \cdot K(x) \cdot\nabla u(x,t) \,dx \\ &+\frac{1}{2}\int_{\Omega}\nabla v(x,t)^{\top} \cdot K(x) \cdot\nabla v(x,t)+ (uv)^2(x,t) \,dx. \end{align*} There exist positive constants \(C\), \(\gamma,\) such that \(E_u(t)\leqslant C e^{-\gamma\,t}E_u(0)\) for all \(t\geqslant T_0\) for all weak solutions to problem (1) provided that \(E_{u,v}(0)\leqslant R\).

MSC:

35L53 Initial-boundary value problems for second-order hyperbolic systems
35B40 Asymptotic behavior of solutions to PDEs
93B07 Observability
35L71 Second-order semilinear hyperbolic equations
35B35 Stability in context of PDEs
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