非线性Hamilton包含与反应扩散系统的同宿解

Homoclinic solutions of Hamiltonian system is a very important research area, which is very active in the field of mathematical research in recent years. We deeply study the homoclinic solutions for smooth and nonsmooth Hamiltonian system, which can greatly promote the development of nonlinear analysis, at the same time, it will greatly influence upon the disciplines of mathematics, physics, mechanics and other relevant theorys and research methods. The purpose of this project is to make comprehensive use of multiple branches of differential equations and nonlinear analysis, including the qualitative theory of differential equations, spectral theory, variational method, critical point theory, set-valued analysis, nonsmooth analysis,topology, differential manifold etc.,to investigate the existence of homoclinic solutions of Hamilton inclusions and reaction-diffusion systems. This project will investigate the Hamiltonian inclusions and reaction-diffusion system via nonsmooth analysis and critical point theory of strongly indefinite functional which was developed recently, develop new tools and new methods for differential inclusions and strongly indefinite noncompact variational problems, discover the mechanism and law of these two kinds of problems for homoclinic solutions, establish criteria for the existence and multiplicity of nontrivial homoclinic solutions and study the asymptotic behavior of solutions by using the relation between critical point theory and dynamical systems. The completion of this project will contribute to the qualitative theory of smooth and nonsmooth dynamical systems.

Hamilton系统的同宿解是一个非常重要的研究方向，是近年来十分活跃的数学研究领域。深入探讨光滑与非光滑Hamilton 系统的同宿解问题，既能有力地推动非线性分析的发展，也将对数学、物理学、力学等学科的相应理论基础和研究方法产生重要的影响。本项目旨在综合运用微分方程和非线性分析的多个分支，包括微分方程定性理论、谱理论、变分方法、临界点理论、集值分析、非光滑分析、拓扑学、微分流形等，来研究Hamilton包含与反应扩散系统，重点是利用非光滑分析与新近建立的强不定泛函的临界点理论研究Hamilton包含与扩散系统的同宿解，发展处理微分包含与强不定非紧变分问题的新工具与新方法，发现这两类问题同宿解的产生机制和规律，利用临界点理论与动力系统的联系，获得非平凡同宿解的存在性及多重性结果并研究其解的渐近行为。本课题的完成将促进光滑与非光滑动力系统定性理论的发展。

Hamilton系统的同宿解是一个非常重要的研究方向，是近年来十分活跃的数学研究领域。深入探讨光滑与非光滑Hamilton 系统的同宿解问题，既能有力地推动非线性分析的发展，也将对数学、物理学、力学等学科的相应理论基础和研究方法产生重要的影响。本项目旨在综合运用微分方程和非线性分析的多个分支，包括微分方程定性理论、谱理论、变分方法、临界点理论、集值分析、非光滑分析、拓扑学、微分流形等，来研究Hamilton包含与反应扩散系统，本项目紧紧围绕Hamilton 系统与反应扩散系统的关健、核心问题：同宿解的存在性、唯一性或多重性；Hamilton 系统具特定性质的同宿解的存在性、唯一性或多重性；Hamilton 系统同宿解的衰减速度估计等展开研究。我们获得了一系列较为深刻的重要成果，并为生物学、物理学和化学等领域出现的微分方程的处理提供了理论依据。项目组成员已发表SCI论文9篇，其中ESI（前1%）高引论文1篇。本课题的的完成将对Hamilton系统与反映扩散方程的理论起到促进作用。