Blow up of solutions of semilinear heat equations in general domains

TL;DR

This paper studies the blow-up behavior of solutions to a nonlinear heat equation in general domains, showing that solutions can blow up for initial data close to a stationary solution, revealing non-star-shaped global solution sets.

Contribution

It extends previous results on blow-up phenomena from symmetric domains to general domains, demonstrating the non-star-shaped nature of global solutions near stationary states.

Findings

Solutions blow up for initial data close to stationary solutions when p is near critical

The set of initial data leading to global solutions is not star-shaped

Blow-up behavior observed in general domains, not just symmetric ones

Abstract

Consider the nonlinear heat equation $v_t -\Delta v= |v|^{p-1} v$ in a bounded smooth domain $\Omega\subset \R^n$ 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.