本文選題：Rosenau方程 + b族方程��； 參考：《電子科技大學(xué)》2017年碩士論文

【摘要】：非線性波動(dòng)方程是一類重要的數(shù)學(xué)模型,經(jīng)常用于描述自然現(xiàn)象,也是非線性數(shù)學(xué)物理最前沿的研究課題之一.因其本身重要的應(yīng)用背景以及非線性帶來(lái)的數(shù)學(xué)上的困難,引起了人們濃厚的研究興趣,具有廣泛的應(yīng)用性和旺盛的生命力.通過(guò)對(duì)非線性波方程的求解和定性分析的研究,有助于人們弄清系統(tǒng)的本質(zhì)特性,極大地推動(dòng)相關(guān)學(xué)科如物理學(xué)、力學(xué)以及工程技術(shù)的發(fā)展.本文研究?jī)深惙蔷�性波動(dòng)方程的Cauchy問(wèn)題:一類是帶阻尼的Rosenau方程;一類是修正的b族方程,分別對(duì)其解的一些性質(zhì)做了研究.主要研究?jī)?nèi)容包含以下幾個(gè)部分:第一章主要介紹兩類非線性波動(dòng)方程的物理背景、研究意義及國(guó)內(nèi)研究狀況與發(fā)展態(tài)勢(shì).第二章研究一類帶阻尼Rosenau方程的Cauchy問(wèn)題.首先利用壓縮映射原理研究證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,最后利用其相應(yīng)線性方程解的衰減估計(jì)來(lái)研究其解的漸近性.第三章討論一類修正的b族方程解的持久性問(wèn)題.首先給出持久性研究的準(zhǔn)備知識(shí),然后通過(guò)對(duì)方程解的估計(jì),證明當(dāng)初值有一定的衰減持續(xù)性時(shí),方程組的解也和初值有同樣的衰減持續(xù)性質(zhì).
[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.
【學(xué)位授予單位】：電子科技大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175.29

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本文編號(hào)：2070488