Global Existence and Blow Up of Solutions in a Cauchy Viscoelastic Problem

Abstract

The Cauchy problem for a viscoelastic equation with nonlinear damping and source terms is considered. We establish the nonexistence result of global solutions with initial energy controlled above by a critical value by modifying the method introduced in a work by Autuori et al. in 2010. Then we establish global existence for arbitrary initial data or the energy in potential well. These improve earlier results in the literatures.