Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks

Abstract

We consider the nonlinear wave equation in a bounded domain with a delay term in the weakly nonlinear internal feedback u″(x, t) − Δxu(x, t) + μ1σ(t)g1(u′(x, t)) + μ2σ(t)g2(u′(x, t − τ(t))) = 0 and prove the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with the Faedo-Galerkin procedure under a certain relation between the weight of the delay term in the feedback, the weight of the term without delay and the speed of the delay. Furthermore, we study the asymptotic behavior of solutions using the multiplier method and some general weighted integral inequalities.