两类非线性波动方程若干问题的研究

发布时间：2018-06-26 13:12

本文选题：Rosenau方程 + b族方程　；参考：《电子科技大学》2017年硕士论文

【摘要】：非线性波动方程是一类重要的数学模型,经常用于描述自然现象,也是非线性数学物理最前沿的研究课题之一.因其本身重要的应用背景以及非线性带来的数学上的困难,引起了人们浓厚的研究兴趣,具有广泛的应用性和旺盛的生命力.通过对非线性波方程的求解和定性分析的研究,有助于人们弄清系统的本质特性,极大地推动相关学科如物理学、力学以及工程技术的发展.本文研究两类非线性波动方程的Cauchy问题:一类是带阻尼的Rosenau方程;一类是修正的b族方程,分别对其解的一些性质做了研究.主要研究内容包含以下几个部分:第一章主要介绍两类非线性波动方程的物理背景、研究意义及国内研究状况与发展态势.第二章研究一类带阻尼Rosenau方程的Cauchy问题.首先利用压缩映射原理研究证明其解的存在唯一性,再利用凹分析的方法得到其解的爆破,最后利用其相应线性方程解的衰减估计来研究其解的渐近性.第三章讨论一类修正的b族方程解的持久性问题.首先给出持久性研究的准备知识,然后通过对方程解的估计,证明当初值有一定的衰减持续性时,方程组的解也和初值有同样的衰减持续性质.
[Abstract]:Nonlinear wave equation is an important mathematical model which is often used to describe natural phenomena and is also one of the most advanced research topics in nonlinear mathematical physics. Because of its important application background and mathematical difficulties brought about by nonlinearity, people are interested in the research and have extensive application and vigorous vitality. Through the study of solving nonlinear wave equations and qualitative analysis, it is helpful for people to understand the essential characteristics of the system and to promote the development of related disciplines such as physics, mechanics and engineering technology. In this paper, we study the Cauchy problem for two kinds of nonlinear wave equations: one is the Rosenau equation with damping and the other is the modified b family equation. The main research contents are as follows: the first chapter mainly introduces the physical background of two kinds of nonlinear wave equations, the significance of the research and the domestic research situation and development trend. In chapter 2, we study the Cauchy problem for a class of damped Rosenau equations. Firstly, the existence and uniqueness of the solution is proved by using the contraction mapping principle, then the blow-up of the solution is obtained by using the concave analysis method. Finally, the asymptotic behavior of the solution is studied by using the decay estimate of the solution of the corresponding linear equation. In chapter 3, we discuss the persistence of solutions for a class of modified b family equations. First, the preparatory knowledge of the persistence study is given, and then by estimating the solution of the equation, it is proved that when the initial value has a certain attenuation persistence, the solution of the equations has the same attenuation persistence property as the initial value.
【学位授予单位】：电子科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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