Medjden, Mohamed; Tatar, Nasser-Eddine On the wave equation with a temporal non-local term. (English) Zbl 1144.35431 Dyn. Syst. Appl. 16, No. 4, 665-672 (2007). 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. Cited in 20 Documents MSC: 35L20 Initial-boundary value problems for second-order hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 45K05 Integro-partial differential equations Keywords:exponential decay; memory term; relaxation function; viscoelasticity PDFBibTeX XMLCite \textit{M. Medjden} and \textit{N.-E. Tatar}, Dyn. Syst. Appl. 16, No. 4, 665--672 (2007; Zbl 1144.35431)