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.
We consider the stochastic heat equation $\partial_t u(t,x) = \Delta u(t,x) + \dot{W}_\alpha(t,x)$ on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where $\dot{W}_\alpha$ is a Gaussian noise that is white in time and whose spatial covariance is the kernel of $(-\Delta)^{-...
This paper is concerned with the parabolic-elliptic chemotaxis system with weakly singular sensitivity and logistic source:~$ u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v^\alpha}\nabla v) +ru-\mu u^2$, $0=\Delta v-v+u,$ under the homogeneous Neumann boundary in a smooth bounded convex domain $\Omega\subset\mathbb{R}^n$ for...
We establish sharp temporal lower bounds for the full spatial $L^2$-norm of energy solutions to $\partial_tu-\Delta u=vu$ on $\mathbb{R}^n$, with $n\geq3$. For global solutions, nonvanishing at a single time implies the lower bound $ce^{-Ct}$ when $v$ is small in the scale-invariant space $L^\infty_tL^{n/2}_x$, and $ce...
We consider the parabolic Anderson model $\partial_t u=\partial_x^2 u+\xi u$ on $\mathbb{R}_+\times\mathbb{R}$ with $u(0,\cdot)\equiv 1$, where $\xi$ is a spatial white noise. We study the long-time behavior of the spatial integral $U(t):=\int_{-L(t)/2}^{L(t)/2}u(t,x)\,dx$, where $L(t)=\exp(\alpha^3 t^3/24)$ with $\alp...
Let $\Omega\subset\mathbb R^2$ be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let $0<\lambda<\lambda_1(\Omega)$, where $\Lambda_1 (\Omega)$ is the first eigenvalue for $-\Delta$ on $\Omega$ under Dirichlet boundary conditions. For every fixed integer $k\g...
M. del Pino, Ignacio A. Guerra, M. Musso· 0 citations
We consider the radially symmetric version of the reaction-diffusion equation $u_t-d\Delta u=f(u)$ with a monostable nonlinearity $f$, viewed as a model for the spreading of a species with population range $r0$ and $h'(t)=-d u_r(t,h(t))/\delta$. For the one-dimensional case ($N=1$), Du \cite{DN} proved that when $\delt...
Hongye Cao, Jing-Yi Cui, Cheng-Zhe Tang et al.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
