【作者】 赵亮；

【导师】 李桂莲；

【作者基本信息】 太原理工大学 ， 应用数学， 2011， 硕士

【摘要】 本文讨论了一类在非线性势力与内应力联合作用下具有耗散项的梁方程初边值问题的弱解、强解的存在唯一性及其渐进性：即对xi的一阶广义导数,β∈C1且α″≥β′(s)≥α’(α′α″为正常数),F(u,u)=f1(u)+f2(u),f1(u)为非线性势力项,f2(u)为粘性阻尼项,且均在有界集上有界,Ω为Rn中一个有界凸区域且具有光滑的边界aΩ,△为Laplace算子,▽为梯度算子,u,u分别表示u对时间t的二阶和一阶偏导数,‖·‖为通常意义下的L2(Ω)中的范数,未知函数u(x,t)为杆在坐标x处的截面于时刻t的位移,u0(x),u1(x)是u(x,t)在时刻t=0时的已知函数,具体研究内容如下：1、本文简单介绍了国内外对非线性梁方程的研究现状。2、本文给出了一些基本的概念和引理。3、利用Galerkin方法证明了(1)-(3)的弱解的存在唯一性。4、利用Galerkin方法证明了(1)-(3)的强解的存在唯一性。5、进一步证明了初边值问题的强解对初始条件的连续依赖性。更多还原

【Abstract】 In the modern application of mathematics,elastic beam equations of the scholars has been one of the important issues, including the beam equation with dissipation highly valued by people.This article discusses a class of nonlinear forces and internal stress i n the joint effect of the beam with dissipation equation with:Where R,α,γis an arbitrary constant,andβ∈C1且α"≥β’(s)≥α’(α’α"is positive constant), F(u,u)=f1(u)+f2(u), where f1(u) is nonlinear force term, f2(u) is viscous damping, which are b ounded in the bounded-set,Ωis abounded convex domain with smooth b oundary (?)Ωin Rn, andΔis Laplace operator,▽is a gradient operator and u,u stand respectively a second order and a first order partial derivatives, what is more,‖·‖is the usual sense norm of the L2(Ω) and unknown functi on u(x,t) is displacement, u0(x),u1(x) are known functions in t=0.Specific contents are as follows:First, this article introduces the domestic and international research on the status of the Nonlinear Beam Equation.Second, the article gives some important concepts and lemmas and some symbols.Third, the existence and uniqueness of weak solution of the equations (1)-(3) are proved by Galerkin method.Fourth, the strong solutionof the equations (1)-(3)are proved by Galerkin method.Fifth, further evidence of the initial boundary value problem of the global solution of the continuous dependence on initial conditions.更多还原