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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 u0 = σφ, that is, there exists σ∗ > 0 such that spreading happens when σ > σ∗, and vanishing happens when σ ≤ σ∗.