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

摘要

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 11 locally uniformly in RR, or to 00 uniformly in the occupying domain. Moreover, we give the sharp threshold when the initial data u0=σ?u0=σ?, that is, there exists σ?>0σ?>0 such that spreading happens when σ>σ?σ>σ?, and vanishing happens when σ≤σ?σ≤σ?.