【作者】 王海英；

【导师】 李桂莲；

【作者基本信息】 太原理工大学 ， 数学， 2015， 硕士

【摘要】 Kirchhoff型微分方程是Kirchhoff在研究弹性弦的自由振动时,提出的非线性数学物理方程,该类型方程在牛顿力学,宇宙物理,血浆问题和弹性理论等诸多领域都有广泛应用,因此研究这类方程具有深刻的实际意义。本文研究下列具有耗散和阻尼项的Kirchhoff型方程在初始条件和边界条件下,整体解的存在性、唯一性,以及整体吸引子的存在性。其中Ω是R2中具有光滑边界aQ的有界开区域,均为正常数,均为给定的函数。非线性函数全文结构如下：第一章主要对Kirchhoff型方程的研究现状进行介绍；第二章对本文用到的函数空间做出说明,给出基本定义和引理；第三章在Sobolev空间中运用Galerkin方法,证明了上述初边值问题第四章整体解的存在性及唯一性；第五章以半群理论为依据，证明了上述初边值问题整体吸引子的存在性；第六章对本文做了全面总结，并提出某些展望。更多还原

【Abstract】 The Kirchhoff type equation was first proposed by Kirchhoff as an existence of the nonlinear wave equation for free vibration of elastic strings. The equation has great applications in many fields, such as non-Newtonian mechanics, cosmology and astrophysics,plasma problems and elasticity theory, so the study of this kinds of equations has a profound practical significance.In the paper we will study the following Kirchhoff equation with dissipating and damping terms In the initial conditions and boundary conditions the existence and uniqueness of the generalized solution and the existence of global attractor for the system. Where Ω denotes an open bounded domain of R2with smooth boundary,) satisfies both of them are given functions, the nonlinear function and for any s≥0,satisfyingThis paper is organized as follows:In the first chapter, we briefly introduce the present research situation of the Kirchhoff equation;In Chapter2, we gave some important concepts, lemmas and explained part of the functions space;In Chapter3, we used Galerkin method to show the existence of the global solution of the problem;In Chapter4, based on semigroup theory, we proved the existence of the whole attractor of the problem;In Chapter5, for this article, we did some summaries and made some prospects for the future of the equation.更多还原