Spreading dynamics of a Fisher-KPP nonlocal diffusion model with a free boundary

This paper concerns the spreading speed and asymptotical behaviors, which was left as an open problem in \cite{LLW22}, of a Fisher-KPP nonlocal diffusion model with a free boundary. Using a new lower solution, we get the exact finite spreading speed of free boundary that is proved to be the asymptotical spreading speed of solution component $u$. Moreover, for the algebraic decay kernels, we derive the rates of accelerated spreading and the corresponding asymptotical behaviors of solution component $u$. Especially, for the level set $E^{\lambda}(t)$ with $\lambda\in(0,u^*)$, we find an interesting propagation phenomenon different from the double free boundary problem.