【作者】 李银玉；

【导师】 张建文；

【作者基本信息】 太原理工大学 ， 应用数学， 2006， 硕士

【摘要】 偏微分方程是数学理论与实际应用之间的一座重要的桥梁。以物理、力学等其它学科中的问题为背景的偏微分方程的研究，不仅是传统应用数学的一个最主要的内容，而且是当代数学中的一个重要组成部分。随着研究的深入，有些原先可用线性偏微分方程作近似处理的问题，也必须考虑非线性项的影响。因此，偏微分方程研究的主体是非线性偏微分方程。对于非线性偏微分方程的定性研究，它的难度大，很难应像线性方程一样用一个统一的方法来加以处理，其研究往往更紧密地结合相应的实际模型。 近年来，关于非线性偏微分方程中非线性双曲型偏微分方程的定性研究，主要以局部解的存在性(整体解可能不存在)、整体解的存在性、正则性及能量衰减估计等为主，但是大多数研究都是在线性边界条件下进行的，非线性边界条件的研究较少。 本文以力学中弯曲与扭转联合作用下的非线性梁模型为背景，建立了一类非线性偏微分方程组，并就非线性边界条件下的情形进行了定性研究，从理论上为这类非线性边界条件下的非线性偏微分方程组的数值研究提供依据。具体内容如下：更多还原

【Abstract】 Partial differential equation is an important bridge between mathematics theory and actual application. The research on partial differential equation based on problem in physics or mechanics and so on, is not only a most important content of traditional applied mathematics but also a significant portion of modern mathematics. With the research progressing, to some problem that can be approximately solved formerly with linear partial differential equation, we must consider the effect of nonlinear factor now. So, the study on partial differential equation focuses on nonlinear partial differential equation. The qualitative analysis to nonlinear partial differential equation is very difficult, and it is a big trouble to deal it in a common way, just as linear equations. When study them, we are apt to combine relevant actual model tightly.In recent years, the qualitative study on nonlinear qualitative partial differential equation mainly focuses on the existence of partial solutions (global solutions don’t exist perhaps), the existence of global solutions, regularity and the estimation on the energy decrease. But these researches mainly based on the condition of linear boundary, rarely on the condition of更多还原