On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$

Giorgio Fusco; Francesco Leonetti; Cristina Pignotti

doi:10.3934/dcds.2017030

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2017, Volume 37, Issue 2: 725-742. Doi: 10.3934/dcds.2017030

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On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$

University of L'Aquila, DISIM, via Vetoio, Coppito, 67010 L'Aquila, Italy

^* Corresponding author: Giorgio Fusco
^* Corresponding author: Giorgio Fusco

Received: June 2015

Revised: November 2015

Published online: November 2016

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Abstract

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $Ω\subset\mathbb{R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric solutions. In dimension $n=2$ an additional asymptotic result is obtained. These results are based on a pointwise estimate obtained for local minimizers of the Allen-Cahn energy.

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Mathematics Subject Classification: Primary:35B40, 35B06;Secondary:35J20.

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References

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