若干非线性波动方程的解的性质和控制问题
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摘要
波动方程是偏微分方程和分布参数控制理论的一个重要的研究内容,对它的研究必将促进偏微分方程和控制理论的进一步发展.本文的研究内容主要有两个.一是应用偏微分方程理论和Sobolev空间理论研究具非线性阻尼和非线性源项的波动方程的解的性质.二是应用黎曼几何方法,结合Carleman估计等方法研究具变系数主部的波动方程的可控性和能量衰减问题.
第一章是引言,主要介绍本文的研究背景,国内外研究现状及本文的主要结果.
第二章主要研究一些弹性振动系统的解的性质.这些性质有解的全局存在性、解的爆破分析、解的非全局存在.第二章第一节研究如下的波动方程：其中,Ω是Rn(n=1,2,3)中的具有光滑边界(?)Ω的有界区域；m,r≥1；fi(·,·)：R2→R2是给定的函数.ρ是满足以下条件的C1类函数：对s>0假设系统具有负初始能量,当函数f1,f2,初值u0,u1,v0,v1和系统中的参数r,m满足适当的条件时,得到了系统解的全局存在和全局不存在的结论.
第二章第二节研究如下的粘弹性波动方程解的性质：其中,Ω是Rn(n=1,2,3)中的具有光滑边界(?)Ω的有界区域；假设系统具有正初始能量,当函数f1,f2,初值u0,u1,v0,v1和系统中的参数r,m,ρ满足适当的条件时,证明了系统的解不能全局存在.
第二章的第三节研究以下非线性梁振动系统其中,a>0,b>0,p>1,m>1.初始条件为假设梁的右端是铰链连接的,即左端带有输入u(t)=(u1(t),u2(t))和输出y(t)=(y1(t),y2(t)),且满足和
通过构造辅助函数,证明了当初值,输入,输出函数分别满足适当的条件时,系统的解在有限时刻爆破和整体存在.
第三章研究论文的第二个主要内容：具变系数主部的波动方程的可控性和能量衰减问题.第三章中Ω是Rn(n≥2)的一个具有光滑边界Γ的有界区域.假设r由Γ0和Γ1两部分组成：Γ0∪Γ1=Γ,Γ0是Γ的非空的相对开集.记v为边界上的外法向量.在Euclidean度量下,记向量场X的散度为div(X).A(x)=(aij(x))是一个n×n的矩阵函数,其中aij=aji.
第三章第一节做一些准备工作,介绍本章的主要研究方法：黎曼几何方法.
第三章第二节研究如下的带边界记忆条件的波动方程当函数f,k和g满足适当的假设时,获得了系统能量指数衰减的结论.
第三章第三节研究如下带非线性边界反馈的波动方程其中是一个已知函数,(?)是u沿着vA=Av的导数.f是定义在Γ1上的连续的非线性的非负函数.在一些假设条件下,我们分别获得了系统能量指数衰减和多项式衰减的结论.
在第三章第四节,令二阶微分算子其中aij=aji是C1类函数且满足研究如下平行耦合的波动方程的可控性问题其中,和w2(x,t)是边界控制函数,α,β分别表示弹性耦合常数和阻尼耦合常数.应用黎曼几何方法和新的Carleman估计得到了系统的能观测性不等式,从而得到了系统的精确能控性.
Wave equations have been one of the important contents of partial differential equa-tion (PDE) and control theory. Studies to wave equations will accelerate the development of PDE and control theory. The main contents of this thesis consist of two parts. The first one is to study the properties of the solutions for the wave equations with nonlinear damp and nonlinear source. We do this by applying PDE theory and Sobolev space the-ory. The second one is to study the controllability and energy decay for wave equations with variable coefficients principal part. We do this by combining Riemannian geometry method and Carleman estimates.
This thesis consists of three chapters.
In Chapter 1, firstly, a survey on the research background and the research advance of the related work are given. Secondly, the main results obtained in this thesis are listed.
Chapter 2 is devoted to the study on the properties of the solutions for some nonlinear vibration systems. The properties conclude global existence, blow-up, nonexistence of global solution.
In Section 1 of Chapter 2, we consider the system of nonlinear wave equations whereΩis a bounded domain with smooth boundary (?)Ωin Rn,n= 1,2,3;m, r≥1; fi(·,·):R2→R2 are given functions to be specified later. Assume that the initial energy is negative. Under some suitable assumptions on the functions f1 and f2, the initial data and the parameters in the equations, the theorems of global existence and nonexistence are proved, respectively.
Section 2 of Chapter 2 is devoted to the study on the viscoelastic wave equations whereΩis a bounded domain with smooth boundary (?)Ωin Rn,n= 1,2,3;m,r≥1; fi(·,·):R2→R2,i= 1,2, are given functions to be specified later. Assume that the initial energy is positive. Under some suitable assumptions on the functions f1,f2,ρ1,ρ2, g1,g2, parameters r,m and the initial data u0,u1,v0,v1, the global nonexistence theorem for solutions is proved.
Section 3 of Chapter 2 deals with the following damped nonlinear beam equation where a>0,b>0, p>1 and m> 1. Suppose that the initial data are given by Suppose that the right end of the beam is hinged, i.e., and at the left the input u(t)= (u1(t),u2(t)) and the output y(t)=(y1(t),y2(t)) are exerted, satisfying and It is proved that, under some conditions, by constructing auxiliary function, the system has global solution and blow-up solution, respectively.
Chapter 3 is devoted to the second main goal:the study to the controllability and energy decay for wave equations with variable coefficients principal part. In Chapter 3,Ωis a bounded domain in Rn with smooth boundaryΓ. It is assumed thatΓconsists of two parts:Γ0 andΓ1,Γ0∪Γ1=Γ, withΓ0 nonempty and relatively open inΓ. Let v denote the outward normal vector field along the boundary.
In Section 1 of Chapter 3, we do some preliminaries:introduce Riemannian geometry method.
Section 2 of Chapter 3 is concerned with exponential decay of the energy of the following problem: Exponential decay of the energy is proved provided that the function f, k and g satisfy some assumptions. In Section 3 of Chapter 3, we investigate energy decay problem of the following wave equation where (?)(x)∈W1,∞(Ω) is a known function, is the derivative of u alongνA= Aν. f is a continuous nonlinear nonnegative function defined onΓ1, and satisfies some assumptions. Under some different assumptions, polynomial decay and exponential decay are obtained, respectively. In Section 4 of Chapter 3, set be a second-order differential operator, with aij= aji of class C1, satisfying for some constant a0>0. We study exact controllability for the following coupled wave equation with variable coefficients principal part where Q=Ω×(0,T],Σ=Γ×(0,T] andΣi=Γi×(0,T],i= 0,1. w1(x, t) and w2(x, t) are control actions on the boundary, andα,βdenote the spring and damper coupling constants, respectively. Applying Riemannian geometry method and new Carleman esti-mates, the observability inequality and exact controllability for the system are obtained.
第一章是引言,主要介绍本文的研究背景,国内外研究现状及本文的主要结果.
第二章主要研究一些弹性振动系统的解的性质.这些性质有解的全局存在性、解的爆破分析、解的非全局存在.第二章第一节研究如下的波动方程：其中,Ω是Rn(n=1,2,3)中的具有光滑边界(?)Ω的有界区域；m,r≥1；fi(·,·)：R2→R2是给定的函数.ρ是满足以下条件的C1类函数：对s>0假设系统具有负初始能量,当函数f1,f2,初值u0,u1,v0,v1和系统中的参数r,m满足适当的条件时,得到了系统解的全局存在和全局不存在的结论.
第二章第二节研究如下的粘弹性波动方程解的性质：其中,Ω是Rn(n=1,2,3)中的具有光滑边界(?)Ω的有界区域；假设系统具有正初始能量,当函数f1,f2,初值u0,u1,v0,v1和系统中的参数r,m,ρ满足适当的条件时,证明了系统的解不能全局存在.
第二章的第三节研究以下非线性梁振动系统其中,a>0,b>0,p>1,m>1.初始条件为假设梁的右端是铰链连接的,即左端带有输入u(t)=(u1(t),u2(t))和输出y(t)=(y1(t),y2(t)),且满足和
通过构造辅助函数,证明了当初值,输入,输出函数分别满足适当的条件时,系统的解在有限时刻爆破和整体存在.
第三章研究论文的第二个主要内容：具变系数主部的波动方程的可控性和能量衰减问题.第三章中Ω是Rn(n≥2)的一个具有光滑边界Γ的有界区域.假设r由Γ0和Γ1两部分组成：Γ0∪Γ1=Γ,Γ0是Γ的非空的相对开集.记v为边界上的外法向量.在Euclidean度量下,记向量场X的散度为div(X).A(x)=(aij(x))是一个n×n的矩阵函数,其中aij=aji.
第三章第一节做一些准备工作,介绍本章的主要研究方法：黎曼几何方法.
第三章第二节研究如下的带边界记忆条件的波动方程当函数f,k和g满足适当的假设时,获得了系统能量指数衰减的结论.
第三章第三节研究如下带非线性边界反馈的波动方程其中是一个已知函数,(?)是u沿着vA=Av的导数.f是定义在Γ1上的连续的非线性的非负函数.在一些假设条件下,我们分别获得了系统能量指数衰减和多项式衰减的结论.
在第三章第四节,令二阶微分算子其中aij=aji是C1类函数且满足研究如下平行耦合的波动方程的可控性问题其中,和w2(x,t)是边界控制函数,α,β分别表示弹性耦合常数和阻尼耦合常数.应用黎曼几何方法和新的Carleman估计得到了系统的能观测性不等式,从而得到了系统的精确能控性.
Wave equations have been one of the important contents of partial differential equa-tion (PDE) and control theory. Studies to wave equations will accelerate the development of PDE and control theory. The main contents of this thesis consist of two parts. The first one is to study the properties of the solutions for the wave equations with nonlinear damp and nonlinear source. We do this by applying PDE theory and Sobolev space the-ory. The second one is to study the controllability and energy decay for wave equations with variable coefficients principal part. We do this by combining Riemannian geometry method and Carleman estimates.
This thesis consists of three chapters.
In Chapter 1, firstly, a survey on the research background and the research advance of the related work are given. Secondly, the main results obtained in this thesis are listed.
Chapter 2 is devoted to the study on the properties of the solutions for some nonlinear vibration systems. The properties conclude global existence, blow-up, nonexistence of global solution.
In Section 1 of Chapter 2, we consider the system of nonlinear wave equations whereΩis a bounded domain with smooth boundary (?)Ωin Rn,n= 1,2,3;m, r≥1; fi(·,·):R2→R2 are given functions to be specified later. Assume that the initial energy is negative. Under some suitable assumptions on the functions f1 and f2, the initial data and the parameters in the equations, the theorems of global existence and nonexistence are proved, respectively.
Section 2 of Chapter 2 is devoted to the study on the viscoelastic wave equations whereΩis a bounded domain with smooth boundary (?)Ωin Rn,n= 1,2,3;m,r≥1; fi(·,·):R2→R2,i= 1,2, are given functions to be specified later. Assume that the initial energy is positive. Under some suitable assumptions on the functions f1,f2,ρ1,ρ2, g1,g2, parameters r,m and the initial data u0,u1,v0,v1, the global nonexistence theorem for solutions is proved.
Section 3 of Chapter 2 deals with the following damped nonlinear beam equation where a>0,b>0, p>1 and m> 1. Suppose that the initial data are given by Suppose that the right end of the beam is hinged, i.e., and at the left the input u(t)= (u1(t),u2(t)) and the output y(t)=(y1(t),y2(t)) are exerted, satisfying and It is proved that, under some conditions, by constructing auxiliary function, the system has global solution and blow-up solution, respectively.
Chapter 3 is devoted to the second main goal:the study to the controllability and energy decay for wave equations with variable coefficients principal part. In Chapter 3,Ωis a bounded domain in Rn with smooth boundaryΓ. It is assumed thatΓconsists of two parts:Γ0 andΓ1,Γ0∪Γ1=Γ, withΓ0 nonempty and relatively open inΓ. Let v denote the outward normal vector field along the boundary.
In Section 1 of Chapter 3, we do some preliminaries:introduce Riemannian geometry method.
Section 2 of Chapter 3 is concerned with exponential decay of the energy of the following problem: Exponential decay of the energy is proved provided that the function f, k and g satisfy some assumptions. In Section 3 of Chapter 3, we investigate energy decay problem of the following wave equation where (?)(x)∈W1,∞(Ω) is a known function, is the derivative of u alongνA= Aν. f is a continuous nonlinear nonnegative function defined onΓ1, and satisfies some assumptions. Under some different assumptions, polynomial decay and exponential decay are obtained, respectively. In Section 4 of Chapter 3, set be a second-order differential operator, with aij= aji of class C1, satisfying for some constant a0>0. We study exact controllability for the following coupled wave equation with variable coefficients principal part where Q=Ω×(0,T],Σ=Γ×(0,T] andΣi=Γi×(0,T],i= 0,1. w1(x, t) and w2(x, t) are control actions on the boundary, andα,βdenote the spring and damper coupling constants, respectively. Applying Riemannian geometry method and new Carleman esti-mates, the observability inequality and exact controllability for the system are obtained.
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