General decay and blow-up results for a class of nonlinear pseudo-parabolic equations with viscoelastic term

Abstract

In this paper, we consider initial boundary value problem of the generalized pseudo-parabolic equation contain viscoelastic term. We establish firstly the local existence of solutions by Banach fixed point theorem. Then we prove blow-up results for solutions when the initial energy is negative or nonnegative but small enough or positive arbitrary high initial energy, respectively. We also establish the lifespan and the blow-up rate for the weak solution via finding the upper bound and the lower bound for the blow-up times and the upper bound and the lower bound for the blow-up rate. For negative energy, we introduce a new method to prove blow-up results with sharper estimate for upper bound for the blow-up times. Finally, we prove the global existence of the solution and a general decay estimate under some suitable assumptions.