Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level

Abstract

In this paper, we study a three-dimensional (3D) viscoelastic wave equation with nonlinear weak damping, supercritical sources and prescribed past history (Formula presented.), (Formula presented.): (Formula presented.) where the relaxation function k is monotone decreasing with (Formula presented.), (Formula presented.) and (Formula presented.). When the source is stronger than dissipations, i.e. (Formula presented.), we obtain some finite time blow-up results with positive initial energy. In particular, we obtain the existence of certain solutions which blow up in finite time for initial data at arbitrary energy level.