Blow up in a semilinear pseudo-parabolic equation with variable exponents

Abstract

In this paper, we consider the following pseudo-parabolic equation with variable exponents: $$\begin{aligned} u_{t}-\Delta u-\Delta u_{t}+\int _{0}^{t}g(t-\tau )\Delta u(x,\tau )d \tau = |u|^{p(x)-2}u. \end{aligned}$$ Under suitable assumptions on the initial datium $$u_{0}$$ , the relaxation function g and the variable exponents p, we prove that any weak solution, with initial data at arbitrary energy level, blows up in finite time.