October  1999, 5(4): 849-870. doi: 10.3934/dcds.1999.5.849

Nonlinear heat equation: the radial case

1. 

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

Received  October 1998 Revised  June 1999 Published  July 1999

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.
Citation: Arthur Ramiandrisoa. Nonlinear heat equation: the radial case. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 849-870. doi: 10.3934/dcds.1999.5.849
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