Exponential Growth of Solution for a Class of Reaction-Diffusion Equations with Memory Term

In this paper, we study an initial boundary value problem, related to nonlinear pseudo-parabolic equations with a memory term, of the form: \(\displaystyle (1+k|u|^{m-2})u_{t}-\Delta u+\int _{0}^{t}g(t-\tau )\Delta u(\tau )d\tau +au=b|u|^{p-2}u, \) subject to Dirichlet boundary conditions. We show the exponential growth of the \(L^{p}\) -norm of the solution to the above problem, when the initial energy is of any sign, using a differential inequality technique.