$L^2$-contraction and asymptotic stability of Cahn--Hilliard fronts

Bongsuk Kwon; Wanyong Shim

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arXiv · 2606.15658

$L^2$-contraction and asymptotic stability of Cahn--Hilliard fronts

Bongsuk Kwon·Wanyong Shim

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Abstract

We study the stability of transition fronts for the one-dimensional Cahn--Hilliard equation. More precisely, we prove that for any H\"older initial datum sufficiently close to a front in the $L^2$ norm, the corresponding solution of the Cahn--Hilliard equation exists globally and converges, up to a dynamical shift, to the front as time tends to infinity. For the proof, we develop an $L^2$-stability framework adapted to Cahn--Hilliard fronts. The key ingredient is a nontrivial second-order Poincar\'e-type inequality that reveals a coercive structure in the indefinite quadratic energy form associated with the linearized operator about the front, once the dynamical shift is taken into account. This yields an $L^2$-contraction estimate and, via a far-field semigroup argument, asymptotic orbital stability of the front in the unweighted $L^2$ topology. The stability analysis requires only the $L^2$-smallness of the initial perturbation; no higher-order smallness, spatial localization, or moment assumptions are imposed.

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Bongsuk Kwon, Wanyong Shim. 2026-06-14. $L^2$-contraction and asymptotic stability of Cahn--Hilliard fronts. https://arxiv.org/abs/2606.15658

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