【作者】 韩伟伟；

【导师】 李大潜；

【作者基本信息】 复旦大学 ， 应用数学， 2008， 博士

【摘要】 本文系统地研究了拟线性双曲组在半有界初始轴上的柯西问题,分别对其经典解的整体存在性、渐近性态（在整体经典解存在的前提下）、破裂现象（生命跨度及奇性形成）等进行了讨论,并举例说明其应用.本文的结构如下:第一章,简要介绍了有关拟线性双曲组柯西问题（包括在整个初始轴和半有界初始轴上）经典解的研究历史、现状和本文的主要结果.为完整起见,在第二章,我们给出了一些预备知识,包括广义标准化坐标、波的分解公式以及本文所要用到的三个重要引理.第三章考虑了非齐次拟线性双曲组在半有界初始轴上柯西问题的整体经典解.在最大（相应地,最小）的特征弱线性退化及非齐次项满足相应于此特征的匹配条件的假设下,对于小而衰减的初值,得到了柯西问题经典解的整体存在唯一性.第四章在第三章的基础上,讨论了拟线性双曲组在半有界初始轴上柯西问题经典解当t→+∞时的渐近性态.证明了当时间t趋于无穷大时,只要初始数据当x→+∞（相应地,x→-∞）时以速率（1+x）^-（1+μ）(相应地,（1—x）^-（1+μ）衰减,柯西问题的经典解就以速率（1+t）^-μ逼近于C¹行波解的组合,其中μ是一个正常数.第五章主要研究拟线性双曲组在半有界初始轴上柯西问题经典解的破裂现象.在最大（相应地,最小）的特征非弱线性退化及非齐次项满足相应于此特征的匹配条件的假设下,对于小而衰减并满足特定条件的初值,我们得到了经典解生命跨度的精确估计及奇性形成机制.第六章将举例说明第三、四、五章所研究的理论的应用.更多还原

【Abstract】 In this Ph.D.thesis,we systematically study the Cauchy problem on a semibounded initial axis for quasilinear hyperbolic systems.The global existence,the asymptotic behavior（based on the global existence）,the blow-up phenomenon（the lifespan and formation of singularities） of classical solutions to the Cauchy problem are discussed respectively.We also give some examples to show the applications of the theory.The structure of the thesis is as follows:A brief introduction on the research history of classical solutions to the Cauchy problem（both on the whole and a semi-bounded initial axis） for quasilinear hyperbolic systems is given in Chapter 1.The main results of this thesis will be also included in this chapter.For the sake of completeness,in Chapter 2,we give some preliminaries,which contain generalized normalized coordinates,John’s formula on the decomposition of waves and three important lemmas to be used in this thesis.In Chapter 3,we consider the global existence of the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.Under the assumptions that the rightmost（resp.leftmost） eigenvalue is weakly linearly degenerate and the inhomogeneous term satisfies the corresponding matching condition,we obtain the global existence and uniqueness of classical solution with small and decaying initial data.Based on Chapter 3,Chapter 4 deals with the asymptotic behavior of classical solution to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.We prove that,when t tends to the infinity,the solution approaches a combination of C^I travelling wave solutions with algebraic rate（1 + t）^-μ,provided the initial data decay with the rate（1 + x）^-（1+μ）(resp.（1 - x）^-（1+μ）) as x tends to +∞（resp.-∞）,whereμis a positive constant.Chapter 5 is devoted to the study of the blow-up phenomenon of classical solutions to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.Under the assumptions that the rightmost（resp.leftmost） eigenvalue is not weakly linearly degenerate and the inhomogeneous term satisfies the corresponding matching condition,for small and decaying initial data satisfying a special condi-tion,we obtain a sharp estimate and the mechanism of formation of singularities of classical solutions.In Chapter 6,we will give some examples to show applications of the theory we study in Chapter 3,Chapter 4 and Chapter 5.更多还原