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

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 $\delta\downarrow 0$.