Existence, scaling, and spectral gap for traveling fronts in the 2D renormalized Allen--Cahn equation

TL;DR

This paper analyzes the existence, asymptotic behavior, and spectral properties of traveling wave fronts in the 2D renormalized Allen--Cahn equation, revealing a spectral gap that grows linearly with the renormalization parameter.

Contribution

It constructs monotone traveling wave solutions, provides their asymptotic description as the regularization parameter tends to zero, and proves the existence of a spectral gap that increases linearly with the renormalization constant.

Findings

Constructed traveling wave solutions connecting renormalized equilibria.

Derived asymptotic profiles and speeds for small regularization parameters.

Established a spectral gap that grows linearly as the regularization parameter decreases.

Abstract

We study the deterministic skeleton of the renormalized stochastic Allen--Cahn equation in spatial dimension $2$. For all sufficiently small regularization parameters $\delta>0$, we construct monotone traveling wave front solutions connecting the renormalized equilibria, derive a small-$\delta$ asymptotic description of their profile and speed, and identify the leading-order contributions. Linearizing about the wave and working in a naturally chosen weighted space, we show that there exists a spectral gap between the symmetry induced eigenvalue $0$ and the rest of the spectrum. The spectral gap grows linearly in the renormalization constant as $\delta\downarrow 0$.