Sharp estimates of solutions of free boundary problems with nonlocal diffusion

This paper concerns a nonlocal diffusion problem with a free boundary. We first give accurate estimates on the longtime behaviors of solution by constructing suitably upper and lower solutions. In particular, for two important kinds of kernel functions, one of which is compactly supported and the other behaves like $|x{|}^{-\gamma }$ with $\gamma \in (1,2]$ near infinity, some sharp estimates on the longtime behaviors and rates of accelerated spreading are obtained. Then the limiting behaviors of the solution pair of a semi-wave problem and asymptotic dynamics of a nonlocal diffusion problem on half space are given, respectively. Finally, we investigate the limiting profiles of this free boundary problem when the expanding coefficient of free boundary converges to $0$ and $\infty$, respectively.