非线性非局部扩散方程的临界指数研究及其应用

In this project，we plan to study critical exponents of nonlinear nonlocal diffusion equations and their applications to some biology models. Nonlocal diffusion equations which come from population migration and Lévy processes, etc, have become an spotlight topic in the field of PDEs. We mainly plan to consider the critical Fujita exponent and the second critical exponent. Although there are a variety of results concerning the critical Fujita exponent of classical diffusion equations, but much less is known for nonlocal equations, because there are substantial difficulties to be overcome for treating nonlocal diffusion problems, due to such as the complicated representation of the fundamental solution involved,and the absence of scaling invariance. The second critical exponent of some nonlinear local diffusion equations, such as p-Laplacian equation, was recently obtained. The results of the critical Fujita exponent can be used to discuss the hair trigger effect in some population dynamics models, which has important practical value. We plan to consider nonlinear nonlocal diffusions with weighted terms or nonlocal source from above three points and study influence of nonlinear terms and nonlocal diffusion on properties of solutions. We feel that the results and methods of the project will provide some reasonable explanation to the original biology model, and enrich the theory of nonlocal diffusion equations in a certain extent.

本项目拟研究非线性非局部扩散方程的临界指数问题，及其在生物学模型上的应用。非局部扩散方程来源于种群迁徙、Lévy过程等，近年来已成为非线性偏微分方程领域的热点问题之一。这里所考虑的是Fujita临界指数和第二临界指数。尽管关于Fujita临界指数问题的研究在局部扩散方程领域取得了丰富的成果，但非局部扩散方程的此类问题进展缓慢，这是因为非局部扩散方程存在自相似性缺失，基本解极其复杂等本质困难。第二临界指数问题，即使在一些局部扩散方程中，也是近几年才得以解决的。而临界指数问题在生物学上的应用，即用来证明某些生物学模型中存在“hair trigger”效应等，是极富实际意义的问题。我们拟从以上三个方面研究具权函数或具非局部源的非局部扩散方程，考虑非线性项及非局部扩散项之间的相互作用对于解的性质的影响。本项目的研究成果和方法将有助于解释相关生物学现象，并将在一定程度上丰富非局部扩散方程理论。

本项目研究几类非线性非局部扩散方程的临界指数问题及其应用。项目组按计划讨论了具一般核扩散函数和幂型源项的非线性非局部扩散方程的第二临界指数，研究了具非局部型源项的快扩散方程的Fujita临界指数及第二临界指数，证明了一类拟线性趋化模型整体解的存在性。讨论了一类受控哈密顿系统的镇定和控制问题。系统地考虑了几类非线性超临界或临界椭圆方程解的存在性，并对不同的权函数，详细讨论了解的渐近行为。