抽象空间中两类非局部发展方程的适定性与稳定性

非局部发展方程是近年来兴起的微分方程热点问题，引起了包括Wolf奖获得者、美国科学院院士Luis A. Caffarelli等的关注，是一个具有重要理论和实际意义的研究课题。然而非局部项的出现给问题紧性的证明带来了很大的困难和挑战。本项目将综合运用算子半群理论、分数阶预解算子理论、抽象空间微分方程理论及非线性分析的理论工具与思想方法，在抽象空间中研究具有非局部初值条件的非自治发展方程和具有卷积项或分数阶微分算子的非线性发展方程这两类非局部发展方程解的存在性、唯一性及解对参数的连续依赖性，在此基础上建立研究非局部发展方程解的稳定性的理论框架，系统研究两类非局部发展方程解的全局稳定性、条件稳定性及指数稳定性。通过本项目研究，不仅有望解决制约非局部发展方程发展的瓶颈问题并为此类问题的研究提供新的思路和方法，而且将进一步推动算子半群、非线性分析和非局部发展方程理论和应用的发展与交叉融合。

Nonlocal evolution equation is a hot research topic of differential equations in recent years, which has attracted the attention of Luis A. Caffarelli, winner of the Wolf prize and a member of the United States National Academy of Sciences. It is a research subject with important theoretical and practical significance. However, the emergence of nonlocal terms brings great difficulties and challenges to the proof of the compactness of problems. By comprehensive applying the theory of operator semigroups, fractional resolvent operators and differential equations in abstract spaces combined with the theoretical tools and thinking methods developed in nonlinear analysis, this project will investigate the existence and uniqueness of solutions as well as continuous dependence of solutions on data for two classes of nonlocal evolution equations in abstract spaces, including non-autonomous evolution equations with nonlocal initial value conditions and nonlinear evolution equations with convolution item or fractional differential operators. On this basis, we will establish some new frameworks and methodology to investigate the stability of solutions for nonlocal evolution equations and systematically study the global stability, conditional stability and exponential stability of solutions for these two classes of nonlocal evolution equations. Through the research of this project, it is expected to not only solve the bottleneck problem restricting the development of nonlocal evolution equations and provide new ideas and methods for the research of such type of problems, but also further promote the development and cross fusion of theory and applications for operator semigroups, nonlinear analysis and nonlocal evolution equations.