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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 $\mathbb{R}^2$

Francesca De Marchis 1, and  Isabella Ianni 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 [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.
Keywords: sign-changing stationary solutions, Semilinear heat equation, finite-time blow-up, asymptotic behavior..
Mathematics Subject Classification: Primary: 35K91, 35B35, 35B44, 35J9.
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
References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Basel-Boston-Berlin, (2007).   Google Scholar

[11]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Basel-Boston-Berlin, (2007).   Google Scholar

[11]

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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