Sharp profiles for periodic logistic equation with nonlocal dispersal

Abstract

In this paper, we study the nonlocal dispersal logistic equation $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=J*u-u+\lambda u-[b(x)q(t)+\delta ]u^p &{}\text {in}\,\bar{\Omega }\times (0,\infty ),\\ u(x,t)=0 &{}\text {in}\,{\mathbb {R}^N\setminus \bar{\Omega }}\times (0,\infty ),\\ u(x,t)=u(x,t+T) &{}\text {in}\,\bar{\Omega }\times [0,\infty ), \end{array}\right. } \end{aligned}$$here $$\Omega \subset \mathbb {R}^N$$ 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 $$\delta \rightarrow 0$$. 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.