Blow-Up Phenomena for a Class of Parabolic or Pseudo-parabolic Equation with Nonlocal Source

Abstract

In this paper, we consider a class of parabolic or pseudo-parabolic equation with nonlocal source term:

$$\begin{aligned} u_t-\nu \triangle u_t-\hbox {div}(\rho (|\nabla u|)^2\nabla u)=u^p(x,t)\int _{\Omega }k(x,y)u^{p+1}(y,t)dy, \end{aligned}$$

where $\nu \ge 0$ and $p>0$. Using some differential inequality techniques, we prove that blow-up cannot occur provided that $q>p$, also, we obtain some finite-time blow-up results and the lifespan of the blow-up solution under some different suitable assumptions on the initial energy. In particular, we prove finite-time blow-up of the solution for the initial data at arbitrary energy level. Furthermore, the lower bound for the blow-up time is determined if blow-up does occur.