位势井理论及其在Kirchhoff系统中的应用

位势井理论是研究非线性发展方程解的动力学性态重要的手段,并且产生了大量有意义的成果,同时也遗留了大量有趣的重要问题.而Kirchhoff方程和方程组是描述弹性弦非线性振动的著名数学物理模型方程.本项目旨在通过对位势井结构展开深入细致的研究,以期在位势井理论的研究上有所突破,并以此作为技术手段分别研究具粘弹性项非线性Kirchhoff系统和非线性项耦合的Kirchhoff系统.本项目研究上述系统整体解的存在性,长时间行为和有限时间爆破的条件以及条件间的关系.更有特色的是,本项目基于初始能量,在不同的能级下(次临界,临界和超临界)全面研究系统解的动力学性态,同时,本项目将揭示位势井结构的相关系数以及初值所满足条件之间的关系,以及对解的性质的影响.

The potential well theory is an important technique to investigate the dynamic behaviour of solutions for nonlinear evolution equations, which brought numerous meaning results and also left many intereting unsolved problems. And also the Kirchhoff equaitons and systems are some famous mathematical physical models for describing the nonlinear elastic string vibration. Our project aims to undertake a deep study on the struction of potential well and try to make a breakthrough on the potential well theory, which will be used to treat the nonlinear Kirchhoff system with viscoelastic terms and some nonliear Kirchhoff syetem with coupled nonliear source terms. Our project considers the conditions on the global existence, long time behaviour and finite time blow up of solutions for above systems and the relationships between these conditions. What is more, our project undertakes a comprehensive study on the dynamic behaviour of solutions regarding the initial enery at three different energy levels(sub-critical energy level, critical energy level and sup-critical energy level). Moreover our project will reveal the relationships between the corresponding coefficients of the struction of potential well and the conditions on the initial data, and also the influences on the properties of solutions.