Finite time blow-up and global solutions for the viscoelastic wave equation with combined power-type nonlinearities

Abstract

The main objective of this paper is to investigate the global existence and nonexistence of solutions to the viscoelastic wave equation with a linear memory term of Boltzmann type, a nonlinear friction damping and a supercritical source term which is a combination of power-type nonlinearities. The global existence of solutions is obtained provided that the energy sink dominates the energy source in an appropriate sense. In more general scenarios, we prove the global existence of solutions if the initial data is taken from a subset of a suitable potential well. Based on the global existence results, the energy decay rate is described in terms of the relaxation kernel as well as the growth order of the damping term. In addition, we also establish the blow-up results in the case that the source is stronger than the dissipative effect. In particular, we prove the finite time blow-up solutions exist at arbitrarily high initial energy.