Finite time blow-up for the semilinear parabolic equation with a memory term

Abstract

This paper is devoted to the study of finite time blow-up for initial-boundary value problem to a class of semilinear parabolic equation under the influence of a linear memory term

$$\begin{aligned} \displaystyle u_{t}-\Delta u+\int _{0}^{t}g(t-s)\Delta u(x,s)ds =|u|^{p-2}u,\quad \text{ in }\ \ \Omega \times (0, T), \end{aligned}$$

where parameter \(p>2\) and \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^{n}, n\ge 1, T\in (0, \infty ]\) is the maximal existence time of solution. By virtue of the concavity method, variational method, an improved potential well method involving time variable t and some new differential inequality techniques, we obtain three finite time blow-up results for the problem under different initial energy levels and suitably assumptions on the relaxation function g, we also derive the upper bound estimation for the blow-up time. Regarding the hypotheses on initial data and function g, our findings improve several existing finite time blow-up conclusions.