Blow-Up of Solutions for a Class of Nonlinear Pseudoparabolic Equations with a Memory Term

Abstract

We consider the nonlinear pseudoparabolic equation with a memory term $u_t-\Delta u-\Delta u_t+{\int }_0^t\lambda \left(t-\tau \right)\Delta u\left(\tau \right)d\tau =\text{d}\text{i}\text{v}\left({\left|\nabla u\right|}^{p-2}u\right)+u^{1+\alpha }$, $x\in \mathrm{\Omega }$, $t>0$, with an initial condition and Dirichlet boundary condition. Under negative initial energy and suitable conditions on p, $\alpha$, and the relaxation function $\lambda (t)$, we prove a finite-time blow-up result by using the concavity method.