Some qualitative properties of weak solution for pseudo-parabolic equation with viscoelastic term and Robin boundary conditions

Abstract

In this paper, we consider the initial boundary value problem of the generalized pseudo-parabolic equation containing viscoelastic terms and associated with Robin conditions. We establish first the local existence of solutions by the standard Galerkin method. 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 by 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 a sharper estimate for the upper bound for the blow-up times. Finally, we prove both the global existence of the solution and the general decay of the energy functions under some restrictions on the initial data.