article

Title: Multiple blowup for nonlinear heat equations at different places and different times

Authors: Noriko Mizoguchi and Juan Luis Vazquez

Issue: Volume 56 (2007), Issue 6, 2859-2886

Abstract:

$We study a blowup problem for the semilinear heat equation \[ u_t = \Delta u + u^p \] posed in $\mathbb{R}^N \times (0,\infty)$, where $p$ is supercritical in the Sobolev sense. A solution $u(x,t)$ is said to blow up at a time $0 < T < +\infty$ if there exists a sequence $t_n \nearrow T$ as $n \to \infty$ such that $|u(\cdotp, t_n)|_{\infty} \to +\infty$ as $n \to \infty$ with supremum norm $| \cdotp |_{\infty}$ in $\mathbb{R}^N$. We establish the following result: for given times $T_1, T_2$ with $0 < T_1 < T_2$ and any small $\varepsilon > 0$, there exist a proper solution $u$ of the problem and times $T_{\varepsilon, i} $ with $|T_{\varepsilon, i} - T_i| < \varepsilon$, $i = 1, 2$, such that $u$ blows up at the origin at $t = T_{\varepsilon, 1}$, becomes a regular solution for $ t \in (T_{\varepsilon, 1}, T_{\varepsilon, 2})$ and blows up again on a sphere at $t = T_{\varepsilon, 2}$.$