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

TL;DR

This paper investigates the behavior of symmetric solutions to the Allen-Cahn equation in unbounded two-dimensional domains, providing existence, exponential estimates, and asymptotic results based on energy minimizers.

Contribution

It establishes existence, uniform exponential bounds, and asymptotic properties of symmetric solutions in unbounded domains, extending understanding of the Allen-Cahn equation in two dimensions.

Findings

Existence of symmetric solutions under certain conditions.

Uniform exponential estimates for these solutions.

Asymptotic behavior characterized in two-dimensional cases.

Abstract

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $\Omega\subset {\bf R}^n.$ 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.