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 utt(x,t)−Δxu(x,t)+∫t0h(t−s)Δxu(x,s)ds+μ1g1(ut(x,t))+μ2g2(ut(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.
International Journal of Dynamical Systems and Differential Equations, 2014 Vol.5 No.1, pp.1 - 26
Published online: 23 Jan 2015 *Full-text access for editors Access for subscribers Purchase this article Comment on this article