Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term

Abstract

© 2017 Informa UK Limited, trading as Taylor & Francis Group In this paper, we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a nonlinear source term (Formula presented.) where (Formula presented.) is a bounded domain in (Formula presented.) ((Formula presented.)) with a Dirichlet boundary condition. Under suitable assumptions on the initial data (Formula presented.) and the relaxation function g, we obtain the global existence and finite time blow-up of solutions with initial data at low energy level (i.e. (Formula presented.)), by using the Galerkin method, the concavity method and an improved potential well method involving time t. We also derive the upper bounds for the blow-up time. Finally, we obtain the existence of solutions which blow up in finite time with initial data at arbitrary energy level.