# American Institute of Mathematical Sciences

March  2015, 35(3): 891-907. doi: 10.3934/dcds.2015.35.891

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

 1 Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientica 1, 00133 Roma, Italy 2 Dipartimento di Matematica e Fisica, Seconda Università di Napoli, V.le Lincoln 5, 81100 Caserta, Italy

Received  February 2014 Revised  May 2014 Published  October 2014

We consider the semilinear heat equation $$\label{problemAbstract}\left\{\begin{array}{ll}v_t-\Delta v= |v|^{p-1}v & \mbox{ in }\Omega\times (0,T)\\ v=0 & \mbox{ on }\partial \Omega\times (0,T)\\ v(0)=v_0 & \mbox{ in }\Omega \end{array}\right.\tag{\mathcal P_p}$$ 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 ($\mathcal P_p$), under which the solution of ($\mathcal P_p$) 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 [5] this phenomenon has been previously 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.
Citation: Francesca De Marchis, Isabella Ianni. Blow up of solutions of semilinear heat equations in non radial domains of $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 891-907. doi: 10.3934/dcds.2015.35.891
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