Existence, scaling, and spectral gap for traveling fronts in the 2D renormalized Allen--Cahn equation
Authors
Gideon Chiusole
Christian Kuehn
Abstract
We study the deterministic skeleton of the renormalized stochastic Allen--Cahn equation in spatial dimension $2$. For all sufficiently small regularization parameters $δ>0$, we construct monotone traveling wave front solutions connecting the renormalized equilibria, derive a small-$δ$ 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 $δ\downarrow 0$.
