非线性动力系统若干理论问题研究
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摘要
本题目来源于国家自然科学基金项目课题“几类非线性数学物理模型方程与抛物方程”(No．10271034)与“高阶发展方程与Schr(?)dinger方程的动力学性态”(No．10871055)．本文利用位势井族方法定性研究几类非线性动力系统，得到相应数学物理模型方程定解问题解的不变集合和真空隔离．并研究了系统的整体适定性和有限时间爆破．特别是得到了整体解存在的最佳条件(门槛结果)．对于特殊问题，得到解的长时间行为．
首先，文章研究一类具有多个非线性源项的波动方程的初边值问题．该问题来源于量子力学中的Klein-Gordon方程．文章研究的非线性情形是考虑若干同性外源影响下系统的整体适定性问题，但并不限定外源的数量．通过建立变分问题的一般结构，得出整体解存在与不存在的充要条件，并在临界情况下讨论了解的存在性条件．对于此问题进一步讨论了若干异性外源项影响下系统的适定问题．文章对于非正定能量通过建立位势井族理论克服能量估计的困难．阐述了位势井族相应泛函的性质，并得到了其与系统内某特殊球的关系．同时对于具有若干异号源项的波动方程讨论了位势井深度函数的性质．文章得到了其整体解存在与不存在的充分必要条件和真空隔离，并讨论了临界条件下的整体解存在性．利用反应扩散方程与波动方程在势能表达式上的相似性，对于非线性反应扩散方程文章得到了与波动方程平行的结论．
接着，文章讨论了具有复杂结构的系统方程．首先研究从粘性塑性微结构模型的弱非线性分析中提出的一类具耗散和应变的四阶波动方程的初边值问题．在系统能量正定条件下系统的坍塌和临界条件下系统的适定性问题一直是热点问题．本文从变分法的基本理论出发针对此类高阶复杂的非线性模型用积分估计，等价模和积分变换等技巧构造了新的位势井．并得到了相应位势井族的性质．同时还得到了位势井族泛函和H_0~2(Ω)空间中球的对应关系．文章首次对正定导数构成的空间建立变分问题，并研究了其与原变分问题的关系．通过正负导数空间的划分，本文指出最小化算子属于正定导数空间的Nehari流形．同时给出了单个位势井深度的精确估计．在此基础上文章得到了此问题解的不变集合与真空隔离．并在正定能量条件下得到整体解存在与有限时间爆破的最佳条件，即门槛结果．而这些结果可以和H_0~2(Ω)空间中球的内部和外部直接对应．对于初始能量大于零，等于零和小于零的情形，文章均证明了系统的爆破．对于临界初始能量，文章全面讨论了整体解的存在性和不存在性，并得到了适合于临界能量的最佳条件，且并不要求初值的正定内积．由此结论可推知对于反应扩散系统和具耗散的波动系统可不要求初值的正定内积．最后对假设条件进行了叙述上的简化和具体化，使得相关结论可以方便地应用于工程实践．同时也给出了具体工程问题的一些具体函数适用于本模型的例子．最后，利用乘子法，对于具有色散耗散项的四阶波动方程的初边值问题和强阻尼非线性波动方程的初边值问题分别得到了其整体解的长时间行为．
进一步，文章研究了一类广义Boussineq方程．Boussineq方程用以描述具有小扰动的长波水波运动，并频繁地被用于浅海或海港区水波运动的模拟中，对于一类广义Boussineq方程的柯西问题，本文针对f(u)=±|u|~p和f(u)=-|u|~(p-1)(p＞1)讨论整体解存在性与不存在性．首先使用Fourier变换得到系统的能量守恒．对于正定能量和非正定能量的情形得到所对应的位势井的相关性质和解的真空隔离现象．而后基于这些性质，文章证明了上述问题整体解存在与有限时间爆破的门槛结果．对于临界初始能量，文章也得到了上述问题整体解存在与有限时间爆破的门槛结果．但不同于具有耗散的波动系统，该门槛结果要求初值具有正定内积．
最后，本文模拟分析了势能控制函数的性质和初值的性态，并且模拟了位势井族深度函数的形态，分析了复杂源项对上述问题的影响．
The research topics of the present thesis are from the research projects sup-ported by the National Natural Science of China,which are“Several classes ofnonlinear physical mathematics model equations and parabolic equations”(No.10271034) and“Dynamics of high order evolution equations and Schr(o|¨)dingerequations”(No.10871055).This work makes qualitative studies on severalnonlinear dynamical systems by using potential wells method.The invariantsets and vacuum isolations of solutions to the relative mathematical physicsproblems are obtained.And the well-posedness and finite time blows up of so-lutions to the systems are studied.Especially the sharp conditions(thresholdresults) for global existence are given.For some special problems,the long timebehaviors are concluded.
First,the thesis studies the initial boundary value problem for a classof nonlinear wave equations with several nonlinear terms,which is a generalform of Klein-Gordon equation from quantum mechanics.The nonlinearityunder consideration reflects the affects of several nonlinear sources to the well-posedness of the system,without any restriction on the number of the sourceterms.By constructing the variational problem this work derives the sufficientand necessary conditions for global existence of the problem and discusses thecritical cases only for global existence.This work considers the several sourceterms with different signs further.The difficulties of energy estimates for non-positive energy are overcame by constructing the potential well theory.Thethesis clarifies some properties of some relative functionals of potential wells,and gives the relations between these functionals and a special ball.Also thedepth function of potential wells is described for this case of nonlinearities withdifferent signs.The sufficient and necessary conditions for global existence and non-existence are shown,also are the vacuum isolation and the global case forcritical condition.For the fact that the nonlinear reaction-diffusion equationpossesses a similar potential energy expression to that of wave equation,thiswork obtains the parallel results to those of wave equations.
Then,this thesis considers the system equation with complex structure.First this thesis focuses on the initial boundary value problem for a class offourth-order wave equations with dissipative and nonlinear strain terms,whicharose from weakly nonlinear analysis of elasto-plastic-microstructure models.For this problem,the collapse for positive energy case and the well-posednessfor critical condition are both primarily concerned problems.Based on the fun-damental theory of variational problem,in aid of the techniques like integrationestimations,equivalent norms,integration substitution and so on,this thesisintroduces the new potential wells for this class of high order complex nonlin-ear models and gets the properties of the potential wells.And the relationsbetween these functionals and a ball in H_0~2(Ω) are pointed out.In this thesis,it is the first time to set the variational problem for the positive derivativesspace and study the relations with the original variational problem.By divid-ing the space according to positive and negative derivatives,it is pointed outthat the minimizer belongs to Nehari manifold of positive derivatives space.Then this work estimates the depth of a single potential well and obtains theinvariant sets of the problem and vacuum isolation.Then the thesis gives thesharp conditions(threshold results) of global existence and finite time blow upof solutions for positive energy.These results parallel the outside and inside ofthe ball in H_0~2(Ω) space.For the cases that initial energy is greater than zero,equal to zero or less than zero,the system blow up is proved under variousassumptions.For the critical energy,a comprehensive consideration is taken for the global existence and non-existence of solutions,and the sharp conditionis obtained.Then some assumptions are simplified and made more concrete inorder to be applied in engineering practices.Also some special valid functionsin engineering problems are given as examples.For the initial boundary valueproblems of a class of fourth order wave equations with dispersive term anddissipative term and a class of strongly damped nonlinear wave equations re-spectively,by using multiplier method the corresponding asymptotic behaviorsof the global solutions are obtained.
Further,the thesis studies a class of generalized Boussinesq equations.Boussinesq-type equations were introduced to describe the motion of waterwave with small-amplitude long waves,which are frequently used in computermodels for the simulation of water waves in shallow seas and harbors.For theCauchy problem of a class of generalized Boussinesq-type equations,the presentthesis discusses the global existence and non-existence of the open problemswith f(u) =±|u|~p and f(u) =-|u|~(p-1)(p＞1).First,by applying Fouriertransformation the energy conservation is obtained.Then for both positiveenergy and non-positive energy the thesis gives some properties for potentialwells and derives the vacuum isolation of solutions.Based on above derivedproperties,this thesis proves the sharp condition of global-in-time existenceand blow up of solutions to above problems.At the critical energy level,thesimilar sharp condition of global well-posedness of above problems is also ob-tained.Differing from the dissipative system,this sharp condition relies on therequirement of positive inner product of initial data.
Finally the present thesis simulates and analyzes the characters of the po-tential energy functionals,the initial data and the depth of potential wells.Alsothe simulations show how the complex source terms affect the above problems.
首先，文章研究一类具有多个非线性源项的波动方程的初边值问题．该问题来源于量子力学中的Klein-Gordon方程．文章研究的非线性情形是考虑若干同性外源影响下系统的整体适定性问题，但并不限定外源的数量．通过建立变分问题的一般结构，得出整体解存在与不存在的充要条件，并在临界情况下讨论了解的存在性条件．对于此问题进一步讨论了若干异性外源项影响下系统的适定问题．文章对于非正定能量通过建立位势井族理论克服能量估计的困难．阐述了位势井族相应泛函的性质，并得到了其与系统内某特殊球的关系．同时对于具有若干异号源项的波动方程讨论了位势井深度函数的性质．文章得到了其整体解存在与不存在的充分必要条件和真空隔离，并讨论了临界条件下的整体解存在性．利用反应扩散方程与波动方程在势能表达式上的相似性，对于非线性反应扩散方程文章得到了与波动方程平行的结论．
接着，文章讨论了具有复杂结构的系统方程．首先研究从粘性塑性微结构模型的弱非线性分析中提出的一类具耗散和应变的四阶波动方程的初边值问题．在系统能量正定条件下系统的坍塌和临界条件下系统的适定性问题一直是热点问题．本文从变分法的基本理论出发针对此类高阶复杂的非线性模型用积分估计，等价模和积分变换等技巧构造了新的位势井．并得到了相应位势井族的性质．同时还得到了位势井族泛函和H_0~2(Ω)空间中球的对应关系．文章首次对正定导数构成的空间建立变分问题，并研究了其与原变分问题的关系．通过正负导数空间的划分，本文指出最小化算子属于正定导数空间的Nehari流形．同时给出了单个位势井深度的精确估计．在此基础上文章得到了此问题解的不变集合与真空隔离．并在正定能量条件下得到整体解存在与有限时间爆破的最佳条件，即门槛结果．而这些结果可以和H_0~2(Ω)空间中球的内部和外部直接对应．对于初始能量大于零，等于零和小于零的情形，文章均证明了系统的爆破．对于临界初始能量，文章全面讨论了整体解的存在性和不存在性，并得到了适合于临界能量的最佳条件，且并不要求初值的正定内积．由此结论可推知对于反应扩散系统和具耗散的波动系统可不要求初值的正定内积．最后对假设条件进行了叙述上的简化和具体化，使得相关结论可以方便地应用于工程实践．同时也给出了具体工程问题的一些具体函数适用于本模型的例子．最后，利用乘子法，对于具有色散耗散项的四阶波动方程的初边值问题和强阻尼非线性波动方程的初边值问题分别得到了其整体解的长时间行为．
进一步，文章研究了一类广义Boussineq方程．Boussineq方程用以描述具有小扰动的长波水波运动，并频繁地被用于浅海或海港区水波运动的模拟中，对于一类广义Boussineq方程的柯西问题，本文针对f(u)=±|u|~p和f(u)=-|u|~(p-1)(p＞1)讨论整体解存在性与不存在性．首先使用Fourier变换得到系统的能量守恒．对于正定能量和非正定能量的情形得到所对应的位势井的相关性质和解的真空隔离现象．而后基于这些性质，文章证明了上述问题整体解存在与有限时间爆破的门槛结果．对于临界初始能量，文章也得到了上述问题整体解存在与有限时间爆破的门槛结果．但不同于具有耗散的波动系统，该门槛结果要求初值具有正定内积．
最后，本文模拟分析了势能控制函数的性质和初值的性态，并且模拟了位势井族深度函数的形态，分析了复杂源项对上述问题的影响．
The research topics of the present thesis are from the research projects sup-ported by the National Natural Science of China,which are“Several classes ofnonlinear physical mathematics model equations and parabolic equations”(No.10271034) and“Dynamics of high order evolution equations and Schr(o|¨)dingerequations”(No.10871055).This work makes qualitative studies on severalnonlinear dynamical systems by using potential wells method.The invariantsets and vacuum isolations of solutions to the relative mathematical physicsproblems are obtained.And the well-posedness and finite time blows up of so-lutions to the systems are studied.Especially the sharp conditions(thresholdresults) for global existence are given.For some special problems,the long timebehaviors are concluded.
First,the thesis studies the initial boundary value problem for a classof nonlinear wave equations with several nonlinear terms,which is a generalform of Klein-Gordon equation from quantum mechanics.The nonlinearityunder consideration reflects the affects of several nonlinear sources to the well-posedness of the system,without any restriction on the number of the sourceterms.By constructing the variational problem this work derives the sufficientand necessary conditions for global existence of the problem and discusses thecritical cases only for global existence.This work considers the several sourceterms with different signs further.The difficulties of energy estimates for non-positive energy are overcame by constructing the potential well theory.Thethesis clarifies some properties of some relative functionals of potential wells,and gives the relations between these functionals and a special ball.Also thedepth function of potential wells is described for this case of nonlinearities withdifferent signs.The sufficient and necessary conditions for global existence and non-existence are shown,also are the vacuum isolation and the global case forcritical condition.For the fact that the nonlinear reaction-diffusion equationpossesses a similar potential energy expression to that of wave equation,thiswork obtains the parallel results to those of wave equations.
Then,this thesis considers the system equation with complex structure.First this thesis focuses on the initial boundary value problem for a class offourth-order wave equations with dissipative and nonlinear strain terms,whicharose from weakly nonlinear analysis of elasto-plastic-microstructure models.For this problem,the collapse for positive energy case and the well-posednessfor critical condition are both primarily concerned problems.Based on the fun-damental theory of variational problem,in aid of the techniques like integrationestimations,equivalent norms,integration substitution and so on,this thesisintroduces the new potential wells for this class of high order complex nonlin-ear models and gets the properties of the potential wells.And the relationsbetween these functionals and a ball in H_0~2(Ω) are pointed out.In this thesis,it is the first time to set the variational problem for the positive derivativesspace and study the relations with the original variational problem.By divid-ing the space according to positive and negative derivatives,it is pointed outthat the minimizer belongs to Nehari manifold of positive derivatives space.Then this work estimates the depth of a single potential well and obtains theinvariant sets of the problem and vacuum isolation.Then the thesis gives thesharp conditions(threshold results) of global existence and finite time blow upof solutions for positive energy.These results parallel the outside and inside ofthe ball in H_0~2(Ω) space.For the cases that initial energy is greater than zero,equal to zero or less than zero,the system blow up is proved under variousassumptions.For the critical energy,a comprehensive consideration is taken for the global existence and non-existence of solutions,and the sharp conditionis obtained.Then some assumptions are simplified and made more concrete inorder to be applied in engineering practices.Also some special valid functionsin engineering problems are given as examples.For the initial boundary valueproblems of a class of fourth order wave equations with dispersive term anddissipative term and a class of strongly damped nonlinear wave equations re-spectively,by using multiplier method the corresponding asymptotic behaviorsof the global solutions are obtained.
Further,the thesis studies a class of generalized Boussinesq equations.Boussinesq-type equations were introduced to describe the motion of waterwave with small-amplitude long waves,which are frequently used in computermodels for the simulation of water waves in shallow seas and harbors.For theCauchy problem of a class of generalized Boussinesq-type equations,the presentthesis discusses the global existence and non-existence of the open problemswith f(u) =±|u|~p and f(u) =-|u|~(p-1)(p＞1).First,by applying Fouriertransformation the energy conservation is obtained.Then for both positiveenergy and non-positive energy the thesis gives some properties for potentialwells and derives the vacuum isolation of solutions.Based on above derivedproperties,this thesis proves the sharp condition of global-in-time existenceand blow up of solutions to above problems.At the critical energy level,thesimilar sharp condition of global well-posedness of above problems is also ob-tained.Differing from the dissipative system,this sharp condition relies on therequirement of positive inner product of initial data.
Finally the present thesis simulates and analyzes the characters of the po-tential energy functionals,the initial data and the depth of potential wells.Alsothe simulations show how the complex source terms affect the above problems.
引文
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