【作者】 杨伟；

【导师】 郝江浩；

【作者基本信息】 山西大学 ， 运筹学与控制论， 2013， 硕士

【摘要】 关于偏微分方程理论的研究有着很长的历史,并且至今偏微分方程的研究是数学领域的研究热点之一.经过前人不断地努力,目前对偏微分方程的研究逐步转向解的定性理论的研究,如：解的局部存在性、唯一性、整体存在性、稳定性、周期性、爆破等性质.之后,由于许多物理、生物、化学学科中的现象可以用偏微分方程和方程组来描述,耦合波方程解的研究在偏微分方程理论研究中逐步受到重视,成为一个重要的研究领域.本文主要讨论了几类耦合波方程解的爆破、整体存在和整体不存在,特别讨论了当系统中取不同的耦合项和阻尼项时,系统的解分别对应有不同的性质.研究的第一个问题是如下一类具有初边值条件的耦合波动方程组解的爆破问题,利用能量法给出了解爆破的充分条件.研究的第二个问题是如下一类具有初边值条件的耦合波动方程组解的整体存在性,其中阻尼项为ut,耦合项为f1(u，v),f2(u,v).利用能量法给出了解整体存在的充分条件.研究的第三个问题是第二个问题中所研究系统的整体解的不存在性,利用能量法给出了整体解不存在的充分条件.更多还原

【Abstract】 The development and research of partial differential equations have a long history and they are still an important part in math. Due to some scholars’continuous efforts now researchers gradually focus on the qualitative theory and the characteristics of solutions of partial differential equation such as uniqueness, stability, periodicity, blow-up, global existence and so on. Because the phenomenon and fact come from physics, biology and chemistry, a larger number of scholars begin to study the solution of coupled wave equation which is a significant area in partial differential equation.This paper mainly discusses the blow-up, global existence and global non-existence of the solution of coupled wave equations.In the first problem, by using energy method we deal with the blow-up of the solution of the following weak coupled wave equation with initial and boundary value problem, And we obtain a sufficient condition for blow-up of the solution.In the second problem, we argue the global existence of the solution for the following problem, where the damping term is ut and coupling terms are f1(u,v), f2(u,v).In the third problem, we further study the global non-existence of the problem which we discuss in the second problem.更多还原