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Nonlinear heat equation: the radial case

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  • 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.
    Mathematics Subject Classification: 35K55.


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