BLOW-UP RATE FOR NON-NEGATIVE SOLUTIONS OF A NON-LINEAR PARABOLIC EQUATION

Abstract

We study solutions of the equation \begin{equation*} \begin{aligned} & u_t = \sum^n_{i,j=1} {\partial \over \partial x_j} \left(a^{ij} {\partial \over \partial x_i} u^m \right) + h u^p \quad\quad {\rm on}\quad \Omega \times(0,T) \\ & u = 0 \quad\quad {\rm on} \quad \partial \Omega \times (0,T),\end{aligned} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, and $a^{ij} = a^{ij}(x)$ is uniformly positive definite and $h = h(x) \gt 0$ on $\Omega$. When $$0 \lt m \lt 1 \lt p \lt m + \frac{2}{n+1} \quad \textrm{or} \quad 1 \lt m \lt p \leq m + \frac{2}{n+1},$$ we will show that if $u$ is a non-nagative solution and blows up at $T$, then $$u(x,t) \leq C |T-t|^{-1/(p-a)}.$$ The proof relies on rescaling arguments and some, old and new, Fujita-type results.