Spectral Theory Of Nonlocal Operators And Applications

This dissertation is mainly devoted to investigating spectral theory of nonlocal operators and applications in the spatial and temporal heterogeneous environmen-t.More precisely,we discuss the principal eigenvalue of non-symmetric nonlocal operator,the generalised principal eigenvalue of nonlocal operator,the generalised principal eigenvalue of time-periodic nonlocal operator with applications to the non-local diffusion KPP equations and also study the principal eigenvalue of matrix-type nonlocal operator with application to the multiple genotype stem cell regeneration model with nonlocal epigenetic transitions.The main results of the dissertation consist of the following four parts.The first part concerns the principal eigenvalue of non-symmetric nonlocal op-erator.We establish the max-min characterization of the principal eigenvalue by the adjoint idea,and discuss the continuous differentiable property,monotone prop-erty and asymptotic limits of the principal eigenvalues with respect to the dispersal rate.Then the corresponding results for nonlocal operator are applied to nonlocal dispersal logistic equation,focusing on the existence,uniqueness and stability of positive steady state solution,the continuous property and asymptotic limits of the positive steady state solution with respect to the dispersal rate and the definition of total biomass of species by the positive steady state solution.Our results show in a heterogeneous environment,the total biomass is always strictly greater than the total carrying capacity in the special case when the nonlocal dispersal is allowed.The second part studies the effects of dispersal distance and dispersal budget on nonlocal diffusion equations with Neumann boundary condition.More precisely,we are mainly concerned with asymptotic behaviors of generalised principal eigenvalue to the nonlocal dispersal operator,positive stationary solutions and solutions to the nonlocal diffusion KPP equation in both large and small dispersal distance.For large dispersal distance,we show that their asymptotic behaviors are unitary with respect to the cost parameter.However,small dispersal distance can lead to different asymptotic behaviors as the cost parameter is in a different range.The third part is concerned with the generalised principal eigenvalue for time-periodic nonlocal dispersal operators.We establish the equivalence between two different characterisations of the generalised principal eigenvalue,and investigate the dependence of the generalised principal eigenvalue on the frequency of the peri-odic oscillation,the dispersal rate and the dispersal spread.Then these qualitative results are applied to time-periodic nonlinear KPP equations with nonlocal disper-sal,focusing on the effects of the frequency of the periodic oscillation,the dispersal rate and the dispersal spread on the existence and stability of positive time-periodic solutions to nonlinear equations.The last part investigates the principal eigenvalue of matrix-type nonlocal op-erator,which involves the existence,monotonicity,sign and asymptotic behavior of the principal eigenvalue.Then using these results discuss the existence,uniqueness and stability of positive steady state solution and the long time asymptotic behav-ior for the multiple genotype stem cell regeneration model with nonlocal epigenetic transitions,analyse the existence and uniqueness of positive steady state solution and establish some calculable criteria in the special case as the mutation constant is small enough.The results show that the long time behaviors of all genotypes stem cell populations are coincident that is extinction,normal survival or abnormal growth together under the proper conditions.