On a thermo-viscoelastic system with variable exponent sources

Abstract

In this paper, we study the following nonlinear thermo-viscoelastic system {u????u+???0g(t?s)?u(s)ds+v=|u|?????�u, {v???v=u^?, in a bounded domain ??R?. By employing the modified potential well theory, we discuss the properties of global existence and finite time blow-up of the solutions of the problem. Moreover, in the former case by constructing a suitable Lyapunov functional we prove the exponential decay for solutions and in the latter one we give the lower and upper bounds for blow-up time of the blow-up solutions.