Sobolev空间框架下基于变分理论的Klein-Gordon-Maxwell系统的研究

In recent years, the Klein-Gordon-Maxwell system has become one of research hotspots in mathematics and physics, because of its rich physical meaning and various application backgrounds. Based on the theory of function spaces and combined with the variational theory of nonlinear functional analysis, this project investigates the standing wave solutions in the framework of special Sobolev space and related problems for Klein-Gordon-Maxwell system in bounded domains. It includes: (1) the research of the existence and multiplicity of non-radial solutions, positive solutions and ground state solutions for Klein-Gordon-Maxwell system in general bounded domains; (2) the research of the existence and multiplicity of radial symmetric solutions and non-radial solutions, existence of positive solutions and ground state solutions for Klein-Gordon-Maxwell system in symmetric bounded domains (including ball and annalus). The research of this project is the enrichment and development of Klein-Gordon-Maxwell system theory, and will provide new theoretical tools for the research of solitary waves of nonlinear wave equations by developing the Sobolev functional space.

Klein-Gordon-Maxwell系统有着丰富的物理意义和广泛的应用背景，是近年来数学和物理学领域的研究热点之一。基于泛函空间理论，结合非线性泛函的变分理论，本项目将研究特定Sobolev空间框架下的Klein-Gordon-Maxwell系统驻波解及相关问题。主要内容有：（1）研究一般有界域上Klein-Gordon-Maxwell系统的非对称解、正解及基态解的存在性和多解性；（2）研究对称有界区域（包括球邻域和环域）上Klein-Gordon-Maxwell系统的径向对称解和非径向对称解的存在性和多解性、正解及基态解的存在性。本项目的研究不仅是Klein-Gordon-Maxwell系统理论的丰富和发展，也将通过进一步发展Sobolev函数空间理论为非线性波动方程组的驻波解的研究提供新的理论工具。

本项目主要研究特定Sobolev空间框架下的Klein-Gordon-Maxwell系统的驻波解、正解等特殊解的存在性及多解性及相关非线性系统（Kirchhoff系统，Boolean Control Networks等）的优化和控制。..本项目获得了如下有意义的研究成果：（1）基于函数空间理论，刻画不同类型的KGM 系统对应的Sobolev 解空间及特殊的解子空间，从而发现和建立KGM 系统的各类驻波解的动力学性态； (2) 对一般有界区域和球邻域，环域等对称有界区域上KGM 系统的各类驻波解的存在性，多解性给出系统的刻画，同时发现这些解之间的内在联系； (3) 针对Kirchhoff系统，Boolean Control Networks等特殊的非线性系统的优化和控制，设计了Boolean Network有限时域最优控制问题的有效的动态规划算法，并证明了一类非线性Kichhoff系统的边界控制的指数稳定性。..本项目的完成为Klein-Gordon-Maxwell系统理论的丰富和发展，也为布尔网络的控制和优化问题等提供新的理论工具和算法。