Blow-up of a Degenerate Non-linear Heat Equation

Abstract

We study the blowup behavior of non-negative solutions of the following problem: \begin{equation*} \begin{array}{rll} u_t & \hspace{-0.2cm} \displaystyle = u^{p}(\Delta u + u^q) & \quad {\rm in}\quad \Omega \times (0,T),\\ u(x,t) & \hspace{-0.2cm} \displaystyle = 0 & \quad {\rm whenever} \quad x \in \partial \Omega, \end{array} \end{equation*} with $p \gt 0$ and $q \gt 1$. We will show that it is possible to have solutions blowing up at only one point, and $$\limsup_{t \to T^-} \left( (T-t)^{1/(p+q-1)} \max_\Omega u(x,t) \right) = \infty.$$