Two-end solutions to the Allen-Cahn equation in $\mathbb{R}^{3}$

TL;DR

This paper constructs solutions to the three-dimensional Allen-Cahn equation with specific growth rates, using compactness and moduli space theory, expanding understanding of solution behaviors in nonlinear PDEs.

Contribution

It proves the existence of solutions with arbitrary growth rates greater than rom the real line, employing novel compactness and moduli space methods.

Findings

Existence of solutions with growth rate k for all k > rom the real line.

Solutions asymptotically resemble a shifted heteroclinic profile.

Development of a moduli space framework for analyzing solution varieties.

Abstract

In this paper we consider the Allen-Cahn equation $$ -\Delta u = u-u^3 \ \mbox{in} \ {\mathbb R}^3 $$ We prove that for each $k\in\left( \sqrt{2},+\infty\right),$ there exists a solution to the equation which has growth rate $k$, i.e. $$ \| u-H(\cdot -k \ln r + c_k) \|_{L^\infty} \to 0$$ The main ingredients of our proof consist: (1) compactness of solutions with growth $k$, (2) moduli space theory of analytical variety of formal dimension one.