Decay and blow-up for a viscoelastic wave equation of variable coefficients with logarithmic nonlinearity

Abstract

In this article, we consider a viscoelastic wave equation of variable coefficients with logarithmic nonlinearity and dynamic boundary conditions in a bounded domain. The existence of a global solution is given by use of the potential well method. In the stable set, with some suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity by use of the Riemannian geometry method, logarithmic Sobolev inequality, and Lyapunov functional method. Meanwhile, in the unstable set, blow-up of the solution is also obtained.