Large-diffusion dynamics for a planar Neumann heat equation with exponential nonlinearity

Abstract: We study the Neumann problem $u_t-\varepsilonΔu=e^u-1-au\ (a&gt;1)$ on a smooth bounded domain $Ω\subset\mathbb{R}<sup>2$. For the spatially homogeneous problem, $0$ is stable, the positive equilibrium $ξ<em>a$ 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 $H<sup>1$ ball, and that blow-up occurs precisely when the spatial mean crosses $ξ_a$. For initial data with $|u_0|</em>{H^1}\le R$ and spatial mean at most $ξ<em>a-δ$, let $\varepsilon</em>{\mathrm{unif}}(R,δ)$ denote the uniform diffusion threshold above which all such solutions are global and converge to $0$. We prove $$\log \varepsilon_{\mathrm{unif}}(R,δ)=R^{2/(8π)+O(\log} R)$$ as $R\to\infty$. 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.