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 $r<h(t)$ and density $u(t,r)$ ($r=|x|$), where the free boundary $r=h(t)$ is governed by $u(t,h(t))=\delta >0$ and $h^{\prime }(t)=-d{u}_r(t,h(t))/\delta$. For the one-dimensional case ($N=1$), Du \cite{DN} proved that when $\delta \in (0,1)$, spreading occurs: $u\to 1$ locally uniformly in $\mathbb{R}$, $h(t)\to \infty$, and ${\lim }_{t\to \infty }[h(t)-c_{\ast }t]=\tilde{h}\in \mathbb{R}$ with no logarithmic shift. In the present paper we consider $N\ge 2$ and establish a complete trichotomy: spreading for $\delta \in (0,1)$; transition for $\delta =1$, where $u\to 1$ uniformly on $[0,h(t)]$ and $h(t)\to h_{\infty }\in (0,\infty )$; and vanishing for $\delta >1$, where $h(t)\to 0$ 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 ${\lim }_{t\to \infty }[h(t)-c_{\ast }t+c_N(\delta )\log t]=\hat{h}\in \mathbb{R}$, with the coefficient $c_N(\delta )>0$ satisfying ${\lim }_{\delta \to 0}{c}_N(\delta )=d(N-1)/c_0$, 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.