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.