| Abstract: |
| This talk is devoted to the investigation of a nonlocal dispersal equation in a bounded and connected domain of RN with smooth boundary, with time-periodic nonlinear function of generalized KPP type. We first study the principal spectral theory of a suitable linearized operator. 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. We next study the influences of the principal eigenvalue on the global dynamics and confirm that the principal eigenvalue being zero is critical. It is then followed by the study of the effects of the dispersal rate D and the dispersal range characterized by sigma on the principal eigenvalue and the positive time-periodic solution, and prove various asymptotic behaviors of the principal eigenvalue and the positive time-periodic solution when D and sigma tend to infinity. To achieve these, we develop new techniques to overcome substantial 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 the time-periodic nonlocal operator. |
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