The periodic principal eigenvalues with applications to the nonlocal dispersal logistic equation

In this paper, we study the nonlocal dispersal equation{ut=∫RNJ(x−y)u(y,t)dy−u+λu−a(x,t)up in Ω¯×(0,+∞),u(x,t)=0 in (RN∖Ω¯)×(0,+∞),u(x,0)=u0(x) in Ω¯, where Ω⊂RN is a bounded domain, λ and p>1 are constants. The dispersal kernel J is nonnegative. The function a∈C(Ω¯×R) is nonnegative and T-periodic in t, but a(x,t) has temporal or spatial degeneracies (a(x,t) vanishes). We first study the periodic nonlocal eigenvalue problems with parameter and establish the asymptotic behavior of principal eigenvalues when the parameter is large. We find that the spatial degeneracy of a(x,t) always guarantees a principal eigenfunction. Then we consider the dynamical behavior of the equation if a(x,t) has temporal or spatial degeneracies. Our results indicate that only the temporal degeneracy can not cause a change of the dynamical behavior, but the spatial degeneracy always causes fundamental changes, whether or not the temporal degeneracy appears.