Nonlocal dispersal equations in time-periodic media: Principal spectral theory, limiting properties and long-time dynamics

Abstract

The present paper is devoted to the investigation of the following nonlocal dispersal equationut(t,x)=Dσm[∫ΩJσ(x−y)u(t,y)dy−u(t,x)]+f(t,x,u(t,x)),t>0,x∈Ω‾, where Ω⊂RN is a bounded and connected domain with smooth boundary, m∈[0,2) is the cost parameter, D>0 is the dispersal rate, σ>0 characterizes the dispersal range, Jσ=1σNJ(⋅σ) is the scaled dispersal kernel, and f is a time-periodic nonlinear function of generalized KPP type. This paper is a continuation of the works of Berestycki et al. [3,4], where f was assumed to be time-independent. We first study the principal spectral theory of the linear operator associated to the linearization of the equation at u≡0. We establish an easily verifiable, general and sharp sufficient condition for the existence of the principal eigenvalue as well as important sup-inf characterizations of the principal eigenvalue. Next, we study the influences of the principal spectrum point on the global dynamics and confirm that the principal spectrum point being zero is critical. It is followed by the investigation of the effects of the dispersal rate D and the dispersal range characterized by σ on the principal spectrum point and the positive time-periodic solution. In particular, we prove various limiting properties of the principal spectrum point and the positive time-periodic solution as D,σ→0+ or ∞. To achieve these, we develop new techniques to overcome fundamental difficulties caused by the lack of the usual L2 variational formula for the principal eigenvalue, the lack of the regularizing effects of the semigroup generated by the nonlocal dispersal operator, and the presence of the time-dependence of the nonlinearity f. Finally, we establish the maximum principle for time-periodic nonlocal dispersal operators.