Blow-up of solutions for an integro-differential equation with a nonlinear source

Abstract

We study the nonlinear viscoelastic wave equation $$ u_{tt}-Delta u+int_0^t g(t-s)Delta u(s)ds=|u|^p u, $$ in a bounded domain, with the initial and Dirichlet boundary conditions. By modifying the method in [15], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of solutions are also given.