Blow-Up, Exponential Grouth of Solution for a Nonlinear Parabolic Equation with p(x) - Laplacian

Abstract

In this paper, we consider the following equation $u_{t}-\func{div}\left( \left\vert \nabla u\right\vert ^{p\left( x\right)-2}\nabla u\right) +\omega \left\vert u\right\vert ^{m\left( x\right)-2}u_{t}=b\left\vert u\right\vert ^{r\left( x\right) -2}u.$ We prove a finite time blowup result for the solutions in the case $\omega =0$ and exponential growth in the case $\omega >0$, with the negative initial energy in the both case.