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

摘要

We consider the viscoelastic wave equation in a bounded domain with a delay term in the non-linear internal feedback u_(tl)(x,t)-Δ_xu(x,t)+∫_0h(t-s)Δ_xu(x,s)ds+μ_1g_1(u_1(x,t))+μ_2g_2(u_1(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.