【作者】 刘建；

【导师】 盛其荣；

【作者基本信息】 新疆大学 ， 应用数学， 2007， 硕士

【摘要】 我们知道：对于非线性波动方程的混合问题而言，即使初值充分光滑，其经典解的整体存在性一般是无法保证的，更不用说其能量的衰减性。本文从一个特殊的齐次线性波动方程的初边值问题入手，使用Galerkin方法论证了该问题的解的存在性，并且用能量估计的方法得到一个能量估计式，以此为基础得到了给定条件下阻尼波动方程的初边值问题的解的存在唯一性及其能量衰减估计，最后，借助于以上结果，通过构造适当的函数空间，定义度量，应用Banach不动点定理，证明了形如的半线性阻尼波动方程混合问题在适当条件下的经典解的存在唯一性及其能量指数衰减性。全文包含五个引理和两个主要定理以及证明过程，第一节是引言部分，从两个物理模型出发引出问题；第二节，先给出两个主要结果；第三节介绍并证明五个引理；第四节是两个主要结果的证明，全文重点在三四节。更多还原

【Abstract】 As far as nonlinear hyperbolic equation be concerned, the existence of itsclassical solution may not be guaranteed, let along the decay of its energy. Thispaper starts with a special initial boundary value problem of homogeneous linear equation, proves the existence of classical solution of the problem by usingthe Galerkin’s method and consequently yields a energy decay estimate by usingenergy method. On the base of above, We studies the uniqueness of classical solutions and the exponential decay of energy of the initial boundary value problemBy the aid of all the results, we define function space and metric, use Banach’s Fixed Point Theorem. Then in a bounded cone_region, we prove theexistence and uniqueness of classical solutions and the exponential decay of energy of the initial boundary value problem for the semi_linear damping waveequationThis paper contents five lemmas and two theorems as well their proves. Thefirst section is introduction which draw forth the problems from two physicalmodels. The second one gives us two principal results. The third one introducesand proves the five lemmas. The fourth is the prove of the two principal results.We will see the emphasis of this paper in the third and fourth sections.更多还原