Asymptotic behavior of solutions of a Fisher equation with free boundaries and nonlocal term

We study the asymptotic behavior of solutions of a Fisher equation with free boundaries and the nonlocal term (an integral convolution in space). This problem can model the spreading of a biological or chemical species, where free boundaries represent the spreading fronts of the species. We give a dichotomy result, that is, the solution either converges to 1 locally uniformly in R, or to 0 uniformly in the occupying domain. Moreover, we give the sharp threshold when the initial data u(0) = sigma phi, that is, there exists sigma* > 0 such that spreading happens when sigma > sigma*, and vanishing happens when sigma <= sigma*.