## Differential and Integral Equations

- Differential Integral Equations
- Volume 14, Number 5 (2001), 589-604.

### The shape of blow-up for a degenerate parabolic equation

Julián Aguirre and Jacques Giacomoni

#### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 14, Number 5 (2001), 589-604.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356123258

**Mathematical Reviews number (MathSciNet)**

MR1824745

**Zentralblatt MATH identifier**

1161.35424

**Subjects**

Primary: 35K65: Degenerate parabolic equations

Secondary: 35B35: Stability 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations

#### Citation

Aguirre, Julián; Giacomoni, Jacques. The shape of blow-up for a degenerate parabolic equation. Differential Integral Equations 14 (2001), no. 5, 589--604. https://projecteuclid.org/euclid.die/1356123258