Differential and Integral Equations

The shape of blow-up for a degenerate parabolic equation

Julián Aguirre and Jacques Giacomoni

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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


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