Blow up of solutions of semilinear heat equations in general domains

Abstract

Consider the nonlinear heat equation $v_t-\Delta v=|v{|}^{p-1}v$ in a bounded smooth domain $\Omega \subset {\mathbb{R}}^n$ with $n>2$ and Dirichlet boundary condition. Given $u_p$ a sign-changing stationary solution fulfilling suitable assumptions, we prove that the solution with initial value $\theta u_p$ blows up in finite time if $|\theta -1|>0$ is sufficiently small and if $p$ is sufficiently close to the critical exponent. Since for $\theta =1$ the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.

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