若干非线性双曲方程解的全局稳定性与爆破问题
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摘要
双曲方程是偏微分方程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理论和其它数学分支的进一步发展.本文的研究内容主要有两个.一是应用势井理论和(?)Sobolev空间理论研究具非线性阻尼和源项及粘弹性项的波动方程的解的爆破.二是应用Lyapunov能量法,结合势井理论研究波动方程的解的全局存在性和能量衰减问题.
论文分为三章.
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双曲系统的解的爆破,解的局部存在以及全局存在性.主要包括粘弹性波动方程的Cauchy问题,边界上带有分数阶耗散项的波动方程,以及各向异性的波动方程.
第二章第一节考虑以下的粘弹性波动方程的柯西问题：其中m≥2,p>2.函数g:R+→R+是G1类函数,满足以下假设初值u0,u1和参数m,p满足以下假设带有紧支集.(G3)当n≥3时,22.假设系统具有负的初始能量,系统中的核函数和参数满足适当的条件时,我们分别得到了解在有限时刻爆破和全局存在的结论.
第二章第二节讨论以下初边值问题其中Ω是Rn(n≥1)中的具有光滑边界aQ的有界区域.边界由两部分组成：(?)Ω=Г0∪Г1,Г0∩Г1=(?),其中r0与r1在(?)Ω上是可测的,带有(n-1)-维Lebesgue测度λn-1(Γi),i=0,1. v是(?)Ω上的单位外法向量.函数f(u)=｜u｜p-2u是多项式源项,p>2.核函数是弱奇次核,其中0<α<1,β,b>0均为常数.系统中的卷积项代表u的分数阶导数(Caputo意义下).假设系统具有正的初始能量,系统中的参数α,β,p满足适当的条件时,我们借助势井理论和凸分析的方法得到了解在有限时刻爆破的结论.
第二章的第三节研究了以下各向异性的波动方程其中pi≥2,i=1,...,n,T>0,Ω是Rn(n≥1)中的有界开子集,带有光滑边界(?)Ω,f(u)=u|u|σ-2,σ>1假设参数pi(i=1,2,…,n),σ满足一定的条件下：我们证明了局部解的存在性和带负初始能量的解的爆破.
第三章研究论文的第二个主要内容：带有非线性阻尼和源项的波动方程解的全局存在性和能量衰减问题.
第三章第一节研究如下的带边界阻尼和源项的粘弹性波动方程这里m≥2,p≥2.Ω是Rn(n≥1)中的有界区域,带有光滑边界(?)Ω,且(?)Ω=Г0∪Г1,Г0∩Г1=(?),其中Γ0和r1在(?)Ω上是可测的,带有(n-1)-维Lebesgue测度λn-1(Γi),i=0,1.v是(?)Ω上的单位外法向量.g是一个正的核函数.当核函数具有一般的衰减性,且与参数满足一定的条件时,我们利用势井理论和Lyapunov能量法得到了全局解的存在性和能量具有与核函数一致衰减率的结论.
第三章第二节研究如下的拟线性波动方程其中Ω是RN中的有界区域,带有光滑边界(?)Ω.
做如下假设：函数
阻尼项具有形式源项为其中参数p满足：当N=1,2时,p≥1;当N≥3时,1≤p≤N/N-2.
非线性应变项σ(s)满足：对任意的s≥0,其中Ai,bi,di(i=1,2)都是非负常数,且b1+b2>0.
利用微分不等式和解的延拓原理,通过讨论非线性应变项,阻尼项,源项的增长阶的关系,我们得到了上述系统整体解存在的几个新的充分条件.
第三章第三节研究如下耦合的非线性波动方程其中m,r≥1,Ω是RN中的有界区域,带有光滑边界(?)Ω.
全文做如下假设：
非线性应变项σ(s)∈C1满足：且对任意的s≥0,其中b1,b2都是非负常数,且b1+b2>0.
源项f1,f2和初值u0,u1,u0,u1满足以下假设：其中其中a,b>0,p≥3.初值满足利用微分不等式和解的延拓原理,通过讨论非线性应变项,阻尼项,源项的增长阶的关系,我们得到了上述耦合系统整体解存在的几个新的充分条件.
Hyperbolic equations is an important contents of partial differential equation (PDE) . Studies to hyperbolic equations will promote the further development of PDE theory and other branches of mathematics. The main contents of this thesis consist of two parts. The first one is to study the blow-up of solutions for wave equations with nonlinear damping and source terms. We do this by applying Potential Well theory and Sobolev space theory. The second one is to study global existence and energy decay for wave equations. We do this by combining Lyapunov energy method and Potential Well theory.
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 solutions for some hyperbolic systems. The properties include local existence, global existence and blow-up of solutions. Hyperbolic systems include Cauchy problem for viscoelastic wave equation, wave equation with a fractional boundary dissipation, and an anisotropic wave equation.
In Section 1 of Chapter 2, we consider the following Cauchy problem with viscoelastic term where m≥2,p≥2.
g:R+→R+is a C1-function satisfying The initial data u0,u1 and the parameters m,p satisfy the following assumptions with compact support.
(G3) 2< p<2n-1/n-2, if n≥3, and p>2, if n=1,2.
Assume that the initial energy is negative. Under some suitable assumptions on the kernel function and the parameters in the equation, we establish a finite-time blow-up result and a global existence result, respectively.
Section 2 of Chapter 2 is devoted to the study on the following wave equation with fractional derivative term on a part of its boundary WhereΩis a bounded open subset of Rn(n≥1), with a smooth boundary (?)Ω. And (?)Ω=Г0∪Г1,Г0∩Г1=(?),whereГ0 andГ1 are measurable over (?)Ω, endowed with the (n-1)-dimensional Lebesgue measureλn-1(Гi),i=0,1. v is the unit outward normal to (?)Ω. The function f(u)=|u|p-2u is a polynomial source, p>2. The function ia a weakly singular kernel, where 0<α<1,β,b>0. The convolution term in the problem represents a modified fractional derivative of u(in the sense of Caputo).
Assume that the initial energy is positive. By using the potential well theory and con-cavity method, we prove that under some suitable assumptions on the parametersα,β,p in the equation, the solution of the problem blows up in finite time.
Section 3 of Chapter 2 deals with the following anisotropic nonlinear hyperbolic equation Where pi≥2, i=1,…,n, T> 0,Ωis a bounded open subset of Rn(n≥1), with a smooth boundary (?)Ω,g(u)=u|u|σ-2, (σ>1) is a polynomial source.
Under some restriction on the parameters and the initial data, we obtain several results on the local existence of the solution and the blow-up of solutions.
Chapter 3 is devoted to studying the second main content:the study on global existence and energy decay of solutions for wave equations with nonlinear damping and source terms.
In Section 1 of Chapter 3, we consider the following viscoelastic wave equation Here m≥2, p≥2.Ωis a bounded open subset of Rn(n≥1), with a smooth boundary (?)Ω=Г0∪Г1,Г0∩Г1=(?), where F0 and F1 are measurable over (?)Ω, endowed with the (n-1)-dimensional Lebesgue measureλn-1(Гi),i=0,1.νis the unit outward normal to (?)Ω.上. g is a positive kernel function.
By using the potential well theory and Lyapunov energy method, we prove that, under some appropriate assumptions on the kernel function and the parameters, the solution of the problem exists globally and the energy has a general decay.
Section 2 of Chapter 3 is concerned with the global existence of solutions for the following quasilinear wave equation: whereΩis a bounded domain of RN. with a smooth boundary (?)Ω.
We make the following assumptions:σis a function which satisfies for s>0,
The nonlinear damping term has the form
The source term has the polynomial form where the parameters p satisfies:
The nonlinear strain termσ(s) satisfies for s≥0, where ai,bi, di,(i=1,2) are nonnegative constants, and b1+b2>0.
By using differential inequality and continuation principle, and analyzing the param-eters in the equation, some new sufficient conditions for global existence of the solution were obtained.
In Section 3 of Chapter 3, we investigate the following coupled quasilinear wave equation where m,r≥1.Ωis a bounded open domain of RN, with a smooth boundary (?)Ω.
We make the following assumptions
σis a C1 function which satisfies for s>0, and for all s≥0, where b1, b2 are nonnegative constants, and b1+b2>0.
The source terms f1, f2 have the form and where a, b>0, p≥3. And the initial data u0, u1, v0,v1 satisfy By using differential inequality and continuation principle, and analyzing the relationship between the growth orders of the nonlinear strain term, the damping term and the source term, some new sufficient conditions for global existence of the solution were obtained.
论文分为三章.
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双曲系统的解的爆破,解的局部存在以及全局存在性.主要包括粘弹性波动方程的Cauchy问题,边界上带有分数阶耗散项的波动方程,以及各向异性的波动方程.
第二章第一节考虑以下的粘弹性波动方程的柯西问题：其中m≥2,p>2.函数g:R+→R+是G1类函数,满足以下假设初值u0,u1和参数m,p满足以下假设带有紧支集.(G3)当n≥3时,2
第二章第二节讨论以下初边值问题其中Ω是Rn(n≥1)中的具有光滑边界aQ的有界区域.边界由两部分组成：(?)Ω=Г0∪Г1,Г0∩Г1=(?),其中r0与r1在(?)Ω上是可测的,带有(n-1)-维Lebesgue测度λn-1(Γi),i=0,1. v是(?)Ω上的单位外法向量.函数f(u)=｜u｜p-2u是多项式源项,p>2.核函数是弱奇次核,其中0<α<1,β,b>0均为常数.系统中的卷积项代表u的分数阶导数(Caputo意义下).假设系统具有正的初始能量,系统中的参数α,β,p满足适当的条件时,我们借助势井理论和凸分析的方法得到了解在有限时刻爆破的结论.
第二章的第三节研究了以下各向异性的波动方程其中pi≥2,i=1,...,n,T>0,Ω是Rn(n≥1)中的有界开子集,带有光滑边界(?)Ω,f(u)=u|u|σ-2,σ>1假设参数pi(i=1,2,…,n),σ满足一定的条件下：我们证明了局部解的存在性和带负初始能量的解的爆破.
第三章研究论文的第二个主要内容：带有非线性阻尼和源项的波动方程解的全局存在性和能量衰减问题.
第三章第一节研究如下的带边界阻尼和源项的粘弹性波动方程这里m≥2,p≥2.Ω是Rn(n≥1)中的有界区域,带有光滑边界(?)Ω,且(?)Ω=Г0∪Г1,Г0∩Г1=(?),其中Γ0和r1在(?)Ω上是可测的,带有(n-1)-维Lebesgue测度λn-1(Γi),i=0,1.v是(?)Ω上的单位外法向量.g是一个正的核函数.当核函数具有一般的衰减性,且与参数满足一定的条件时,我们利用势井理论和Lyapunov能量法得到了全局解的存在性和能量具有与核函数一致衰减率的结论.
第三章第二节研究如下的拟线性波动方程其中Ω是RN中的有界区域,带有光滑边界(?)Ω.
做如下假设：函数
阻尼项具有形式源项为其中参数p满足：当N=1,2时,p≥1;当N≥3时,1≤p≤N/N-2.
非线性应变项σ(s)满足：对任意的s≥0,其中Ai,bi,di(i=1,2)都是非负常数,且b1+b2>0.
利用微分不等式和解的延拓原理,通过讨论非线性应变项,阻尼项,源项的增长阶的关系,我们得到了上述系统整体解存在的几个新的充分条件.
第三章第三节研究如下耦合的非线性波动方程其中m,r≥1,Ω是RN中的有界区域,带有光滑边界(?)Ω.
全文做如下假设：
非线性应变项σ(s)∈C1满足：且对任意的s≥0,其中b1,b2都是非负常数,且b1+b2>0.
源项f1,f2和初值u0,u1,u0,u1满足以下假设：其中其中a,b>0,p≥3.初值满足利用微分不等式和解的延拓原理,通过讨论非线性应变项,阻尼项,源项的增长阶的关系,我们得到了上述耦合系统整体解存在的几个新的充分条件.
Hyperbolic equations is an important contents of partial differential equation (PDE) . Studies to hyperbolic equations will promote the further development of PDE theory and other branches of mathematics. The main contents of this thesis consist of two parts. The first one is to study the blow-up of solutions for wave equations with nonlinear damping and source terms. We do this by applying Potential Well theory and Sobolev space theory. The second one is to study global existence and energy decay for wave equations. We do this by combining Lyapunov energy method and Potential Well theory.
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 solutions for some hyperbolic systems. The properties include local existence, global existence and blow-up of solutions. Hyperbolic systems include Cauchy problem for viscoelastic wave equation, wave equation with a fractional boundary dissipation, and an anisotropic wave equation.
In Section 1 of Chapter 2, we consider the following Cauchy problem with viscoelastic term where m≥2,p≥2.
g:R+→R+is a C1-function satisfying The initial data u0,u1 and the parameters m,p satisfy the following assumptions with compact support.
(G3) 2< p<2n-1/n-2, if n≥3, and p>2, if n=1,2.
Assume that the initial energy is negative. Under some suitable assumptions on the kernel function and the parameters in the equation, we establish a finite-time blow-up result and a global existence result, respectively.
Section 2 of Chapter 2 is devoted to the study on the following wave equation with fractional derivative term on a part of its boundary WhereΩis a bounded open subset of Rn(n≥1), with a smooth boundary (?)Ω. And (?)Ω=Г0∪Г1,Г0∩Г1=(?),whereГ0 andГ1 are measurable over (?)Ω, endowed with the (n-1)-dimensional Lebesgue measureλn-1(Гi),i=0,1. v is the unit outward normal to (?)Ω. The function f(u)=|u|p-2u is a polynomial source, p>2. The function ia a weakly singular kernel, where 0<α<1,β,b>0. The convolution term in the problem represents a modified fractional derivative of u(in the sense of Caputo).
Assume that the initial energy is positive. By using the potential well theory and con-cavity method, we prove that under some suitable assumptions on the parametersα,β,p in the equation, the solution of the problem blows up in finite time.
Section 3 of Chapter 2 deals with the following anisotropic nonlinear hyperbolic equation Where pi≥2, i=1,…,n, T> 0,Ωis a bounded open subset of Rn(n≥1), with a smooth boundary (?)Ω,g(u)=u|u|σ-2, (σ>1) is a polynomial source.
Under some restriction on the parameters and the initial data, we obtain several results on the local existence of the solution and the blow-up of solutions.
Chapter 3 is devoted to studying the second main content:the study on global existence and energy decay of solutions for wave equations with nonlinear damping and source terms.
In Section 1 of Chapter 3, we consider the following viscoelastic wave equation Here m≥2, p≥2.Ωis a bounded open subset of Rn(n≥1), with a smooth boundary (?)Ω=Г0∪Г1,Г0∩Г1=(?), where F0 and F1 are measurable over (?)Ω, endowed with the (n-1)-dimensional Lebesgue measureλn-1(Гi),i=0,1.νis the unit outward normal to (?)Ω.上. g is a positive kernel function.
By using the potential well theory and Lyapunov energy method, we prove that, under some appropriate assumptions on the kernel function and the parameters, the solution of the problem exists globally and the energy has a general decay.
Section 2 of Chapter 3 is concerned with the global existence of solutions for the following quasilinear wave equation: whereΩis a bounded domain of RN. with a smooth boundary (?)Ω.
We make the following assumptions:σis a function which satisfies for s>0,
The nonlinear damping term has the form
The source term has the polynomial form where the parameters p satisfies:
The nonlinear strain termσ(s) satisfies for s≥0, where ai,bi, di,(i=1,2) are nonnegative constants, and b1+b2>0.
By using differential inequality and continuation principle, and analyzing the param-eters in the equation, some new sufficient conditions for global existence of the solution were obtained.
In Section 3 of Chapter 3, we investigate the following coupled quasilinear wave equation where m,r≥1.Ωis a bounded open domain of RN, with a smooth boundary (?)Ω.
We make the following assumptions
σis a C1 function which satisfies for s>0, and for all s≥0, where b1, b2 are nonnegative constants, and b1+b2>0.
The source terms f1, f2 have the form and where a, b>0, p≥3. And the initial data u0, u1, v0,v1 satisfy By using differential inequality and continuation principle, and analyzing the relationship between the growth orders of the nonlinear strain term, the damping term and the source term, some new sufficient conditions for global existence of the solution were obtained.
引文
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