Existence and asymptotic behaviour of solutions for the wave equation with a time varying delay and logarithmic nonlinearity

Abstract

This paper presents the study of a wave equation with a time varying delay and logarithmic nonlinearity utt − Δu + μ1σ(t)g1(ut) + μ2σ(t)g2(ut(x, t − τ (t))) = uln|u|k, in Ω × (0,∞). By using the Galerkin approximation method and potential-well method, the local and global existence of solutions under the conditions of some properties of specific functions as well as the weight of dissipation and delay are obtained. Moreover, by means of Lyapunov function technique and the properties of convex function, we investigated the asymptotic behaviour of solutions. Especially, the energy decay results we obtained are optimal and can be displayed graphically by selecting suitable functional in dissipation term. By considering the interaction of time varying delay and the logarithmic source term, these results extend the earlier ones in the literature [5] and [14].