Blow up of solutions of semilinear heat equations in non radial domains of ${\mathbb{R}}^2$

Abstract

We consider the semilinear heat equation \begin{equation}\label{problemAbstract}\left\{\begin{array}{ll}v_t-Δv= |v|^{p-1}v & \mbox{in}Ω\times (0,T)\\ v=0 & \mbox{on}\partial Ω\times (0,T)\\ v(0)=v_0 & \mbox{in}Ω\end{array}\right.\tag{${\mathcal{P}}_p$} \end{equation} where $p>1$, $\Omega$ is a smooth bounded domain of ${\mathbb{R}}^2$, $T\in (0,+\infty ]$ and $v_0$ belongs to a suitable space. We give general conditions for a family $u_p$ of sign-changing stationary solutions of \eqref{problemAbstract}, under which the solution of \eqref{problemAbstract} with initial value $v_0=\lambda u_p$ blows up in finite time if $|\lambda -1|>0$ is sufficiently small and $p$ is sufficiently large. Since for $\lambda =1$ the solution is global, this shows that, in general, the set of the initial conditions for which the solution is global is not star-shaped with respect to the origin. In previous paper by Dickstein, Pacella and Sciunzi this phenomenon has already been observed in the case when the domain is a ball and the sign changing stationary solution is radially symmetric. Our conditions are more general and we provide examples of stationary solutions $u_p$ which are not radial and exhibit the same behavior.