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Decay rates for solutions of a system of wave equations with memory. (English) Zbl 1010.35012
The author considers the initial-boundary value problem for the system \[ u_{tt}-\Delta u+\int^t_0 g_1(t-s)\Delta u(s)ds+ \alpha\cdot(u-v)=0\quad\text{in }\Omega \times(0,\infty) \]
\[ v_{tt}-\Delta v+\int^t_0 g_2(t-s)\Delta v(s) ds-\alpha\cdot(u-v)=0\quad\text{in }\Omega\times(0,\infty) \] with homogeneous Dirichlet boundary condition, where \(\Omega\subset\mathbb{R}^n\) is a bounded domain. By using Galerkin approximations, it is proved the existence of a unique strong solution (in appropriate Sobolev spaces). Further, by using a Lyapunov functional, it is shown that the solution decays exponentially to zero provided \(g_i\) decays exponentially to zero. If \(g_i\) decays polynomially to zero then the solution also decays polynomially with the same rate of decay.

35B40 Asymptotic behavior of solutions to PDEs
35L55 Higher-order hyperbolic systems
35L20 Initial-boundary value problems for second-order hyperbolic equations
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