A free boundary problem with nonlocal diffusion and unbounded initial range

We consider a free boundary problem with nonlocal diffusion and unbounded initial range, which can be used to model the propagation phenomenon of an invasion species whose habitat is interval $$(-\infty ,h(t))$$ h(t) representing spreading front. Since spatial scale unbounded, different method from existing works about employed obtain well-posedness. Then we prove that always spreads successfully, very bounded range. also show there finite speed if only threshold condition satisfied by kernel function. Moreover, rate accelerated accurate estimates on longtime behaviors solution are derived.