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On the wave equation with a temporal non-local term. (English) Zbl 1144.35431

Summary: This paper is concerned with the asymptotic behavior for an integro-differential equation \[ u_{tt}+ au_t=\Delta u- \int^t_0 h(t- s)\Delta u(s)\,ds,\quad\text{in }\Omega\times\mathbb{R}_+, \]
\[ u= 0,\quad\text{on }\Gamma\times\mathbb{R}_+, \]
\[ u(x, 0)= u_0(x),\quad u_t(x, 0)= u_1(x),\quad\text{in }\Omega, \] which appears in viscoelasticity. It is proved that the energy of the system decays exponentially to zero as time goes to infinity provided that the kernel in the memory term is also exponentially decaying. New assumptions are discussed.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
45K05 Integro-partial differential equations
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