The shape of blow-up for a degenerate parabolic equation

Abstract

In this paper we study the degenerate parabolic problem \[ \begin{cases} u_t+a\,x\cdot\nabla u-|x|^2\Delta u=f(u), & x\in \mathbb R^N,\quad t>0,\\ u(x,0)=u_0(x)\ge0, & x\in \mathbb R^N, \end{cases} \] where $a\in \mathbb R$ and $f: \mathbb R\to \mathbb v$ is a $C^1$ function. We obtain local existence results and then focus on the blow-up behavior when $f$ is such that \[ f(u)>0\text{ and }\int_u^\infty\frac{ds}{f(s)} <\infty\quad\forall u>0. \] In particular, we describe the blow-up time and rate of the nonlocal solutions under quite general conditions. Differences with the corresponding problem with uniform diffusivity (the heat equation) are stressed.