Nonlinear heat equation: the radial case

Arthur Ramiandrisoa

doi:10.3934/dcds.1999.5.849

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1999, Volume 5, Issue 4: 849-870. Doi: 10.3934/dcds.1999.5.849

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Nonlinear heat equation: the radial case

Arthur Ramiandrisoa^1,

1.
Laboratoire d'Analyse Numérique URA CNRS 189, Université Pierre et Marie Curie, 4, place Jussieu 75252 PARIS

Received: October 1998

Revised: June 1999

Published: October 1999

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Abstract

We study here the blow-up set of the maximal classical solution of $u_t -\Delta u = g(u)$ on a ball of $\mathbb R^N$, $N \geq 2$ for a large class of nonlinearities $g$, with $u(x,0) = u_0(|x|)$. Numerical experiments show the interesting behaviour of the blow-up set in respect of $u_0$. As a theoretical background to the method used in this work, we prove an important monotonicity property, that is for a fixed positive radius $r_0$, when the solution gets large enough at a certain time $t_0$, then $u$ is monotone increasing at $r_0$ after $t_0$. Finally, a single radius blow-up property is proved for some large initial conditions.

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Mathematics Subject Classification: 35K55.

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