带指数非线性项的平面Neumann热方程的大扩散动力学
Large-diffusion dynamics for a planar Neumann heat equation with exponential nonlinearity
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中文总结 AI 辅助
本文研究带指数非线性项的平面Neumann热方程的大扩散动力学，证明足够大的扩散可恢复标量三分性，确定了使特定初值解全局收敛的一致扩散阈值的渐近行为。
中文摘要 AI 辅助
我们研究二维欧氏空间中光滑有界区域Ω上的Neumann问题：$u_t - \varepsilon\Delta u = e^u - 1 - au\\ (a>1)$。对于空间齐次问题，0是稳定的，正平衡点$\xi_a$是不稳定的，且从$\xi_a$上方出发的解会在有限时间内爆破。尽管有限时间爆破在任意扩散率下都存在，我们证明足够大的扩散能在每个有界$H^1$球上一致恢复这种标量三分性，且爆破恰好发生在空间均值超过$\xi_a$时。对于满足$\\|u_0\\|_{H^1}\le R$且空间均值不超过$\xi_a-\delta$的初值，令$\varepsilon_{\mathrm{unif}}(R,\delta)$表示所有这类解为全局解且收敛到0的一致扩散阈值，我们证明当$R\to\infty$时，$\log \varepsilon_{\mathrm{unif}}(R,\delta)=R^2/(8\pi)+O(\log R)$。与区域无关的系数$1/(8\pi)$来自尖锐的均值零Moser-Trudinger不等式，匹配的下界通过边界集中的Moser剖面结合局部Kaplan论证得到。
英文摘要
We study the Neumann problem $u_t-\varepsilonΔu=e^u-1-au\ (a>1)$ on a smooth bounded domain $Ω\subset\mathbb{R}^2$. 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 $\|u_0\|_{H^1}\le R$ and spatial mean at most $ξ_a-δ$, let $\varepsilon_{\mathrm{unif}}(R,δ)$ 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.
