Title: Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the non-linear internal feedback

Authors: Abbes Benaissa; Aissa Benguessoum; Salim A. Messaoudi

Addresses: Laboratory of Analysis and Control of PDE, Djillali Liabes University, P. O. Box 89, Sidi Bel Abbès 22000, Algeria ' Laboratory of Analysis and Control of PDE, Djillali Liabes University, P. O. Box 89, Sidi Bel Abbès 22000, Algeria ' Department of Mathematics and Statistics, KFUPM, Dhahran 31261, Saudi Arabia

Abstract: We consider the viscoelastic wave equation in a bounded domain with a delay term in the non-linear internal feedback u_tt(x,t)−Δ_xu(x,t)+∫^t₀h(t−s)Δ_xu(x,s)ds+μ₁g₁(u_t(x,t))+μ₂g₂(u_t(x,t−τ))=0 and prove a global existence result using the energy method combined with the Faedo-Galerkin procedure under assumption of a relation between the weight of the delay term in the feedback and the weight of the term without delay. Furthermore, we study the asymptotic behaviour of solutions using a perturbed energy method.

Keywords: nonlinear wave equations; delay term; decay rate; multiplier method; energy decay; viscoelastic wave equation; nonlinear internal feedback; asymptotic behaviour; perturbed energy; Galerkin approximation.

DOI: 10.1504/IJDSDE.2014.067080

International Journal of Dynamical Systems and Differential Equations, 2014 Vol.5 No.1, pp.1 - 26

Published online: 29 Apr 2015 *

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