    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 \begin{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{$\mathcal P_p$} \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 ($\mathcal P_p$), 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  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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##### References:
  Adimurthi and M. Grossi, Asymptotic estimates for a two dimensional problem with polynomial nonlinearity,, Proc. Amer. Math. Soc., 132 (2004), 1013.  doi: 10.1090/S0002-9939-03-07301-5.  Google Scholar  T. Cazenave, F. Dickstein and F. B. Weissler, Sign-changing stationary solutions and blow up for the nonlinear heat equation in a ball,, Math Ann., 344 (2009), 431.  doi: 10.1007/s00208-008-0312-6.  Google Scholar  F. De Marchis, I. Ianni and F. Pacella, Sign changing solutions of Lane Emden problems with interior nodal line and semilinear heat equations,, Journal of Differential Equations, 254 (2013), 3596.  doi: 10.1016/j.jde.2013.01.037.  Google Scholar  F. De Marchis, I. Ianni and F. Pacella, Asymptotic analysis and sign changing bubble towers for Lane-Emden problems,, accepted for publication in Journal of the European Mathematical Society, ().   Google Scholar  F. Dickstein, F. Pacella and B. Sciunzi, Sign-changing stationary solutions and blow up for the nonlinear heat equation in dimension two,, Journal of Evolution Equation, 14 (2014), 617.  doi: 10.1007/s00028-014-0230-x.  Google Scholar  P. Esposito, M. Musso and A. Pistoia, On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity,, Proc. Lond. Math. Soc., 94 (2007), 497.  doi: 10.1112/plms/pdl020.  Google Scholar  M. Grossi, C. Grumiau and F. Pacella, Lane Emden problems: Asymptotic behavior of low energy nodal solutions,, Ann. Inst. H. Poincare Anal. Non Lineaire, 30 (2013), 121.  doi: 10.1016/j.anihpc.2012.06.005.  Google Scholar  M. Grossi, C. Grumiau and F. Pacella, Lane Emden problems with large exponents and singular Liouville equations,, Journal Math. Pure and Appl., 101 (2014), 735.  doi: 10.1016/j.matpur.2013.06.011.  Google Scholar  V. Marino, F. Pacella and B. Sciunzi, Blow-up of solutions of semilinear heat equations in general domains,, Comm. Cont. Math., 14 (2014), 617.  doi: 10.1142/S0219199713500429. Google Scholar  P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Basel-Boston-Berlin, (2007). Google Scholar  X. Ren and J. Wei, Single-point condensation and least-energy solutions,, Proc. Amer. Math. Soc., 124 (1996), 111.  doi: 10.1090/S0002-9939-96-03156-5.  Google Scholar
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