非局部特征值问题及其应用

Many nonlocal dispersal equations are induced in the study of population ecology, image processing, material science and so on, which has been recently paid great attention for many scholars such as H.Berestycki、J.Rossi、P.Bates、Y.Lou and so on. We are concerned with the principal eigenvalues of nonlocal eigenvalue problems and their applications. Due to the deficiency of regularity and the lack of compactness of nonlocal operator, there are many essential difficulties. The classical theory in PDE may not be used directly to consider nonlocal dispersal problem, and we need to develop new methods. First, we construct the essential theory of nonlocal eigenvalue problems, such as the existence of principal eigenvalues, the dependence on the weight function and the domain and so on. Then, we consider the existence, uniqueness and stability of nonlocal dispersal Lotka-Voltera competition system with spatial degeneracys and nonlocal dispersal SIS epidemic model with spatial inhomogenous. For the higher dimensional nonlocal dispersal epidemic models, we construct the threshold theory of traveling waves. Meanwhile, we will discuss the effect of nonlocal dispersal to the biological populations, and intend to give a good characterization of the nonlocal dispersal equations.

近年来，在种群生态学, 图像处理, 材料科学等学科的研究中，导出了许多非局部扩散方程并引起了国内外著名学者如H.Berestycki、J.Rossi、P.Bates、Y.Lou等的极大兴趣。由于非局部算子紧性与正则性理论的缺失，很多经典方法不能直接应用于相应的非局部问题，这与椭圆问题存在本质区别。本项目致力于研究非局部算子的特征值理论及其应用。主要研究非局部加权特征值问题主特征值的存在性理论及重要性质，进一步研究空间退化非局部扩散L-V竞争系统稳态解的存在唯一性及稳定性，空间异性非局部扩散SIS传染病模型的动力学行为，以及建立高维非局部扩散传染病模型行波解的稳定性并发展相应的方法。同时，希望通过讨论非局部扩散对生物种群的影响，从动力学的角度理解它们的本质特征。

在种群生态学, 图像处理, 材料科学等学科的研究中，导出了许多非局部扩散方程并引起了国内外著名学者的极大兴趣。由于非局部算子紧性与正则性理论的缺失，很多经典方法不能直接应用于相应的非局部问题，这与椭圆问题存在本质区别。本项目致力于研究非局部算子的特征值理论及其应用。主要研究非局部加权特征值问题主特征值的存在性理论及重要性质，进一步研究空间异性非局部扩散SIS传染病模型的动力学行为，以及建立高维非局部扩散传染病模型行波解的存在性与不存在性，并发展了相应的方法。同时，我们通过讨论非局部扩散对生物种群的影响，从动力学的角度理解它们的本质特征。