Energy decay of a viscoelastic wave equation with supercritical nonlinearities

This paper presents a study of the asymptotic behavior of the solutions for the history value problem of a viscoelastic wave equation which features a fading memory term as well as a supercritical source term and a frictional damping term: {u(tt) - k(0)Delta u - integral(infinity)(0) k' (s)Delta u(t - s)ds + vertical bar u(t)vertical bar(m-1) u(t) = vertical bar u vertical bar|(p-1)u, in Omega x (0, T), u(x, t) = u(0)(x, t), in Omega x (-infinity, 0], where Omega is a bounded domain in R-3 with a Dirichlet boundary condition and u(0) represents the history value. A suitable notion of a potential well is introduced for the system, and global existence of solutions is justified, provided that the history value is taken from a subset of the potential well. Also, uniform energy decay rate is obtained which depends on the relaxation kernel - k'(s) as well as the growth rate of the damping term. This manuscript complements our previous work (Guo et al. in J Differ Equ 257:3778-3812, 2014, J Differ Equ 262:1956-1979, 2017) where Hadamard well-posedness and the singularity formulation have been studied for the system. It is worth stressing the special features of the model, namely the source term here has a supercritical growth rate and the memory term accounts to the full past history that goes back to - infinity .