Abstract
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-posedness. 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.
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