Large-diffusion dynamics for a planar Neumann heat equation with exponential nonlinearity
Abstract
We study the Neumann problem ut−εΔu=eu−1−au (a>1) on a smooth bounded domain Ω⊂ℝ2. For the spatially homogeneous problem, $0$ is stable, the positive equilibrium ξa is unstable, and solutions starting above ξa blow up in finite time. Although finite-time blow-up persists at every diffusivity, we show that sufficiently large diffusion recovers this scalar trichotomy uniformly on every bounded H1 ball, and that blow-up occurs precisely when the spatial mean crosses ξa. For initial data with |u0|H1≤ R and spatial mean at most ξa−δ, let εunif(R,δ) denote the uniform diffusion threshold above which all such solutions are global and converge to $0$. We prove log εunif(R,δ)=R2/(8π)+O(log R) as R→∞. The domain-independent coefficient 1/(8π) arises from the sharp mean-zero Moser--Trudinger inequality. A matching lower bound is obtained from boundary-concentrating Moser profiles via a localized Kaplan argument.
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BibTeX
@misc{seo2026largediffusion,
title = {{Large-diffusion dynamics for a planar Neumann heat equation with exponential nonlinearity}},
author = {Juneyoung Seo},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28061},
eprint = {2608.28061},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28061},
}
RIS
TY - UNPB
TI - Large-diffusion dynamics for a planar Neumann heat equation with exponential nonlinearity
AU - Juneyoung Seo
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28061
UR - https://arxiv.org/abs/2608.28061
AB - We study the Neumann problem u_t−εΔu=e^u−1−au (a>1) on a smooth bounded domain Ω⊂ℝ^2. For the spatially homogeneous problem, $0$ is stable, the positive equilibrium ξ_a is unstable, and solutions starting above ξ_a blow up in finite time. Although finite-time blow-up persists at every diffusivity, we show that sufficiently large diffusion recovers this scalar trichotomy uniformly on every bounded H^1 ball, and that blow-up occurs precisely when the spatial mean crosses ξ_a. For initial data with |u_0|_H^1≤ R and spatial mean at most ξ_a−δ, let ε_unif(R,δ) denote the uniform diffusion threshold above which all such solutions are global and converge to $0$. We prove log ε_unif(R,δ)=R^2/(8π)+O(log R) as R→∞. The domain-independent coefficient 1/(8π) arises from the sharp mean-zero Moser--Trudinger inequality. A matching lower bound is obtained from boundary-concentrating Moser profiles via a localized Kaplan argument.
ER -