Sharp profiles for periodic logistic equation with nonlocal dispersal

In this paper, we study the nonlocal dispersal logistic equation ut=J*u-u+lambda u-[b(x)q(t)+delta]upin omega over bar x(0,infinity),u(x,t)=0inRN\omega over bar x(0,infinity),u(x,t)=u(x,t+T)in omega over bar x[0,infinity),here omega subset of RN\documentclass[12pt]{minimal} is a bounded domain, J is a nonnegative dispersal kernel, p>1, lambda is a fixed parameter and delta>0. The coefficients b, q are nonnegative and continuous functions, and q is periodic in t. We are concerned with the asymptotic profiles of positive solutions as delta -> 0. We obtain that the temporal degeneracy of q does not make a change of profiles, but the spatial degeneracy of b makes a large change. We find that the sharp profiles are different from the classical reaction-diffusion equations. The investigation in this paper shows that the periodic profile has two different blow-up speeds and the sharp profile is time periodic in domain without spatial degeneracy.