Global existence and bounds for blow‐up time in a class of nonlinear pseudo‐parabolic equations with a memory term

Abstract

In this article, we consider the initial boundary value problem for a class of nonlinear pseudo‐parabolic equations with a memory term: ut−Δu−aΔut+∫0tg(t−s)Δu(x,s)ds+bu=normalΔpu+k(t)|ufalse|q−2u. Under suitable assumptions, we obtain the local and global existence of the solution by Galerkin method. We prove finite‐time blow‐up of the solution for initial data at arbitrary energy level and obtain upper bounds for blow‐up time by using the concavity method. In addition, by means of differential inequality technique, we obtain a lower bound for blow‐up time of the solution if blow‐up occurs.