Precise propagation rates of the fractional Fisher-KPP equation with free boundaries

Abstract

In this paper, we determine the propagation rates of the Fisher-KPP equation with free boundaries and fractional diffusion $(-\Delta)^s$ $(0<s<1)$, which was considered in \cite{CDN26} as a model for species spreading, where the density function $u(t,x)$ satisfies the fractional Fisher-KPP equation $u_t+(-\Delta)^su=f(u)$ over a one-dimensional spatial domain $(g(t), h(t))$, representing the expanding population range, with $x=g(t)$ and $x=h(t)$ the free boundaries. It was shown in \cite{CDN26} that the model is well-posed, and its long-time dynamics exhibits a spreading-vanishing dichotomy. We show in this paper that, when spreading occurs, \[ \begin{cases} \lim_{t\to\infty}\frac{h(t)}{t^{1/(2s)}} =\lim_{t\to\infty}\frac{-g(t)}{t^{1/(2s)}} =\left(\frac{2^{1-2s}\mu}{1-2s}\right)^{1/(2s)}&\mbox{ if $s\in(0,1/2)$,}\\[2mm] \lim_{t\to\infty}\frac{h(t)}{t\ln t} =\lim_{t\to\infty}\frac{-g(t)}{t\ln t} =\mu&\mbox{ if $s=1/2$,}\\[2mm] \lim_{t\to\infty}\frac{h(t)}{t} =\lim_{t\to\infty}\frac{-g(t)}{t} =c_0(s)&\mbox{ if $s\in(1/2,1)$}, \end{cases} \] where $c_0(s)\in (0,\infty)$ is uniquely determined by the associated semi-wave problem. The level sets of $u(t,x)$ propagate with the same rates as the free boundaries. The proof uses delicate upper and lower solutions constructed from a smooth profile function in the case $s\in (0, 1/2]$, and from the semi-wave profile when $s\in (1/2, 1)$.