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arXiv · 2512.14245

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

Gideon Chiusole·Christian Kuehn
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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 $\delta\downarrow 0$.

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Gideon Chiusole, Christian Kuehn. 2025-12-16. Existence, scaling, and spectral gap for traveling fronts in the 2D renormalized Allen--Cahn equation. https://arxiv.org/abs/2512.14245

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