Large-diffusion dynamics for a planar Neumann heat equation with exponential nonlinearity
We study the Neumann problem $u_t-\varepsilon\Delta u=e^u-1-au\ (a>1)$ on a smooth bounded domain $\Omega\subset\mathbb{R}^2$. For the spatially homogeneous problem, $0$ is stable, the positive equilibrium $\xi_a$ is unstable, and solutions starting above $\xi_a$ 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 $\xi_a$. For initial data with $\|u_0\|_{H^1}\le R$ and spatial mean at most $\xi_a-\delta$, let $\varepsilon_{\mathrm{unif}}(R,\delta)$ denote the uniform diffusion threshold above which all such solutions are global and converge to $0$. We prove $\log \varepsilon_{\mathrm{unif}}(R,\delta)=R^2/(8\pi)+O(\log R)$ as $R\to\infty$. The domain-independent coefficient $1/(8\pi)$ 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.