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arXiv · 2512.00764

Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation

Myeongju Chae·Soyeun Jung
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Abstract

Within the framework developed in \cite{Gr, JLL, RT1}, we rigorously establish the nonlinear instability of roll solutions to the two-dimensional generalized Swift-Hohenberg equation (gSHE). Our analysis is based on spectral information near the maximally unstable Bloch mode, combined with precise semigroup estimates. We construct a certain class of small initial perturbations that grow in time and cause the solution to deviate from the underlying roll solution within a finite time. This result provides a clear transition from spectral to nonlinear instability in a genuinely two-dimensional setting, where the Bloch parameter $\sigma$ ranges over an unbounded domain.

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Myeongju Chae, Soyeun Jung. 2025-11-30. Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation. https://arxiv.org/abs/2512.00764

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