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 the interval $(-\infty,h(t))$ with $h(t)$ representing the spreading front. Since the spatial scale is unbounded, a different method from the existing works about nonlocal diffusion problem with free boundary is employed to obtain the well-posdeness. Then we prove that the species always spreads successfully, which is very different from the free boundary problem with bounded range. We also show that there is a finite spreading speed if and only if a threshold condition is satisfied by the kernel function. Moreover, the rate of accelerated spreading and accurate estimates on longtime behaviors of solution are derived.
AMS Subject Classification (2000): 35K57, 35R09, 35R20, 35R35, 92D25