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

In this paper, we determine the propagation rates of the Fisher-KPP equation with free boundaries and fractional diffusion $(-\mathrm{\Delta }{)}^s$ $(0<s<1)$, 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 $u_t+(-\mathrm{\Delta }{)}^{s}u=f(u)$ 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}μ}{1-2s}\right)^{1/(2s)} & 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} =μ& if $s=1/2$,\\[2mm] \lim_{t\to\infty}\frac{h(t)}{t} =\lim_{t\to\infty}\frac{-g(t)}{t} =c_0(s) & if $s\in(1/2,1)$, \end{cases} where $c_0(s)\in (0,\mathrm{\infty })$ 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 $s\in (0,1\mathrm{/}2]$, and from the semi-wave profile when $s\in (1\mathrm{/}2,1)$.