Blow-up for degenerate nonlinear parabolic problem

Abstract

In this paper, we deal with the existence, uniqueness, and finite time blow-up of the solution to the degenerate nonlinear parabolic problem: $u_{\tau}=\left( \xi^{r}u^{m}u_{\xi}\right) _{\xi}/\xi^{r}+u^{p}$ for$\;0 < \xi < a$,$\;0 < \tau < \Gamma$, $u\left( \xi,0\right) =u_{0}\left( \xi\right) $ for $0\leq\xi\leq a$, and $u\left( 0,\tau\right) =0=u\left( a,\tau\right) $ for $0 < \tau < \Gamma$, where $u_{0}\left( \xi\right) $ is a positive function and $u_{0}\left( 0\right) =0=u_{0}\left( a\right) $. In addition, we prove that $u$ exists globally if $a$ is small through constructing a global-exist upper solution, and $u_{\tau}$ blows up in a finite time.