基尔霍夫方程的解

In recent years, study on nonlocal partial differential equations has been paid extensive attention. A class of nonlocal partial differential equations related with string vibration and film vibration in physics will be studied in the present project. This kind of equations is usually called Kirchhoff equation in literatures. Kirchhoff equation is nonlocal wave equation, and its steady equation is elliptic Kirchhoff equation. By using methods in nonlinear functional analysis such as critical point theory and bifurcation theory, we firstly study the existence and dependence on parameter of solution to elliptic Kirchhoff equations, which can make us deeply understand the role of nonlocal term. And then we can obtain some valid methods and ideas to deal with nonlocal equations. On this basis, referencing good research methods and ideas about one-dimensional wave equations, we consider the existence of periodic solution to one-dimensional Kirchhoff equation and then obtain some elementary research results regards as Kirchhoff equations.

近年来，非局部偏微分方程的研究受到广泛关注。本项目研究物理学中与弦振动、薄膜振动有关的一类非局部偏微分方程，文献中经常称之为Kirchhoff方程。Kirchhoff方程是非局部波动方程，其稳态方程是椭圆型Kirchhoff方程。利用临界点理论、分歧理论等非线性泛函分析方法，我们首先研究椭圆型Kirchhoff方程解的存在性以及解对某些参数的依赖特性，加深对非局部的认识，获得一些处理非局部项方程的可行的研究方法和思路。以此基础，借鉴研究一维波动方程的好的思路和方法，我们讨论一维Kirchhoff方程周期解的存在性，争取获得一些Kirchhoff方程研究的初步结果。

非局部偏微分方程是一类重要的非线性方程。项目组主要讨论了两类椭圆型Kirchhoff问题，一类是有界区域椭圆型Kirchhoff方程，一类是全空间椭圆型Kirchhoff方程。对于有界区域椭圆型Kirchhoff问题，项目组利用变分方法构造了Kirchhoff方程的Fucik谱中的三条曲线，并研究了这些曲线的分析性质和渐近行为。Kirchhoff方程的Fucik谱可用来研究带有跳跃反应项Kirchhoff问题解的存在性。项目组还讨论了有界区域Kirchhoff型传输问题基态解的存在性以及解的光滑性。对于全空间椭圆型Kirchhoff方程，项目组主要讨论了Kirchhoff非局部项和卷积产生的非局部项的区别以及它们对解的适定性的影响。项目完成了预期目标，提升了项目组成员的科研水平，加深了对Kirchhoff方程及相关问题的认识，为进一步研究打下坚实的基础。