分数阶Klein-Gordon问题和相关变分法的研究

This proposal intends to study the problems about Klein-Gordon fields in quantum field theory. We mainly consider Klein-Gordon systems and Klein-Gordon-Schrödinger systems, including fractional cases and critical cases. The fractional cases in our proposal are the fractional Laplacian acting on the spatial variables and the Caputo fractional derivative acting on the time variable, as well as their synthesis...We intend to find new nonlinear analysis methods and techniques related to partial differential equations to solve the Klein-Gordon problems including fractional cases. The purpose of the proposal is establishing the existence, uniqueness, multiplicity and correlation properties of solutions to the Klein-Gordon problems, including solitary wave solutions, standing wave solutions, traveling wave solutions and ground state solutions, investigating the well posedness of solutions to Cauchy problems, and obtaining what conditions the initial value meets, the solutions will blow up or globally exist. Among the tools involved, we mention (nonlocal) concentration compactness principle, minimax priciple, index theory and methods related to partial differential equation etc.

本项目拟研究量子场论中有关Klein-Gordon场的问题，主要考虑Klein-Gordon方程组和Klein-Gordon-Schrödinger方程组，包括分数阶情形和临界情形。本项目中的分数阶指的是作用在空间变量上的分数阶Laplacian和作用在时间变量上的Caputo分数阶导数以及二者的综合。..我们拟寻找新的非线性分析方法和偏微分方程有关技巧，使之适用于解决分数阶情形的Klein-Gordon问题，建立Klein-Gordon相关问题解的存在性、惟一性、多重性和相关性态，包括孤立波解、驻波解、行波解、基态解等，得到Cauchy问题解的适定性，初值满足什么条件时解爆破、散射和整体存在。非局部集中紧方法、极大极小原理、指标理论以及偏微分方程相关方法等将会被涉及到。

本项目研究了量子场论中有关Klein-Gordon场的问题，主要考虑Klein-Gordon方程组和Klein-Gordon-Schrödinger方程组，包括分数阶情形和临界情形。本项目中的分数阶指的是作用在空间变量上的分数阶Laplacian和作用在时间变量上的Caputo分数阶导数以及二者的综合。..本项目主要研究了由Klein-Gordon方程及Born-Infeld理论耦合的非线性系统的孤波解的存在性，研究了带有临界非线性项的分数阶系统的基态解的存在性，研究了带有磁场和临界指数的分数阶Choquard方程的基态解的存在性，研究了带有临界指数的分数阶Schrödinger方程的基态解的存在性，研究了分数阶双变量Hardy-Sobolev不等式及其最佳常数的可达性问题以及带有磁场的情形。