Differential and Integral Equations

The shape of blow-up for a degenerate parabolic equation

Julián Aguirre and Jacques Giacomoni

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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

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

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K65: Degenerate parabolic equations
Secondary: 35B35: Stability 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations


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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