The high dimensional monostable reaction-diffusion equation with free boundary and radial symmetry

Abstract

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 $\delta\in(0,1)$, spreading occurs: $u\to1$ locally uniformly in $\mathbb{R}$, $h(t)\to\infty$, and $\lim_{t\to\infty}[h(t)-c_*t]=\tilde{h}\in\mathbb{R}$ with no logarithmic shift. In the present paper we consider $N\ge2$ and establish a complete trichotomy: spreading for $\delta\in(0,1)$; transition for $\delta=1$, where $u\to1$ uniformly on $[0,h(t)]$ and $h(t)\to h_\infty\in(0,\infty)$; and vanishing for $\delta>1$, where $h(t)\to0$ and $u\to\delta$ uniformly on $[0,h(t)]$. For the spreading regime, by constructing sharp upper and lower solutions, we prove that the solution converges globally to the semi-wave profile and reveal a logarithmic shift of the form $ \lim_{t\to\infty}\big[h(t)-c_*t+c_N(\delta)\log t\big]=\hat{h}\in\mathbb{R}$, with the coefficient $c_N(\delta)>0$ satisfying $ \lim_{\delta\to0}c_N(\delta)=d(N-1)/c_0$, where $d(N-1)/c_0$ is the shift coefficient for the high-dimensional radial pushed-case Cauchy problem. These results reveal the connection to the spreading behavior modeled by the corresponding Cauchy problem.