The criterion for the existence of principal eigenvalues of time periodic some nonlocal reaction diffusion operators and applications

Abstract

Using several new notions of the generalized principal eigenvalue, we exhibit three distinct sufficient conditions for the existence of the principal eigenfunction of a nonlocal dispersal operator and apply the established theory to study the behavior of the solution of the corresponding nonlocal dispersal equations in the literature. Relations between these generalized principal eigenvalues, the dependence of their different parameters, and several properties of nonlocal operators are further explored. Our approach is to construct several new generalized principal eigenvalues and use new calculations techniques to overcome the fundamental difficulties caused by the unsuitable Harnack-type inequality for elliptic-type nonlocal dispersal equations [33] and the presence of the time dependence of the nonlinear function. Mathematics Subject Classification: 35K57; 47G20; 92D25.