几类随机微分方程的渐近行为
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摘要
动力系统是研究系统演化规律的数学学科.现实中许多系统会不可避免地受到随机因素的影响,把随机因素引入到动力系统中就产生了随机动力系统.随机动力系统出现在许多重要的实际应用领域中,如物理学、力学、海洋学、气象学、生物学、通讯工程及其它的科学与工程技术.
动力系统理论中最基本的、最重要的研究课题之一就是研究动力系统的渐近行为.随机吸引子是描述随机动力系统的渐近行为的有效工具.本文主要研究由带有Brown运动的几类随机微分方程生成的随机动力系统以及随机时滞格点动力系统的渐近行为,考虑系统的随机吸引子的存在性问题.这不仅具有重要的理论意义,而且具有重要的应用意义.
全文分为三个部分:
第一部分是本文的前两章,首先,第一章阐述本文的研究背景和现状以及本文的研究内容和意义.然后,第二章介绍与本文相关的一些基础知识.
第二部分,是本论文的核心内容,包括第三、四、五、六章.首先,第三章研究具有齐次Neumann边界条件的随机强阻尼波动方程的渐近行为,研究由此方程生成的随机动力系统的随机吸引子的存在性.其次,第四、五章分别研究定义在无界区域上的带有可乘噪声的随机阻尼波动方程和随机反应扩散方程的渐近行为,考虑由方程生成的随机动力系统的随机吸引子的存在性.最后,第六章研究随机时滞格点动力系统的渐近行为,首先给出在可分Banach空间C ([ν,0], lρp)上随机动力系统存在随机吸引子的充分条件,然后把所得的抽象结果应用到带有随机耦合系数和可加噪声的一阶随机时滞格点系统,证明此系统存在随机吸引子.
第三部分,即本文的第七章,本章对全文进行总结,并提出今后有待进一步研究的一些问题.
Dynamical system is the mathematical science which study on theevolution of systems. Inreality, many systems will inevitably be affectedby random factors. Considering the random factors in the dynamicalsystem will produce random dynamical system. Random dynamicalsystems appear in many important practical applications, such asphysics, mechanics, oceanography, meteorology, biology,communications engineering, other science and engineeringtechnology.
One of the most basic and important research topics of dynamicalsystems is to study the asymptotic behavior of dynamical systems.Random attractor is an effective tool to describe the asymptoticbehavior of random dynamical systems. This paper is devoted to studythe asymptotic behavior of random dynamical systems generated byseveral classes of stochastic differential equations driven by Brownmovement, as well as the asymptotic behavior of stochastic retardedlattice dynamical systems, consider the existence of random attractorsof these systems. This is important not only in theory, but also inapplication.
This paper is divided into three parts:
The first part is the first two chapters. First of all, chapter1introducesthe background and status of this study, as well as the content andsignificance of this study. Then, chapter2describes some basicknowledge related to this paper. The second part is the core content ofthis paper, including the chapters3,4,5and6. Firstly, chapter3isdevoted to study the asymptotic behavior of the stochastic stronglydamped wave equation with homogeneous Neumann boundarycondition. We investigate the existence of a random attractor for therandom dynamical system associated with the equation. Secondly, chapter4and chapter5study the asymptotic behavior of stochasticdamped wave equation and the stochastic reaction–diffusionequation with multiplicative noise on unbounded domain, and considerthe existence of random attractors for the random dynamical systemsgenerated by the two equations, respectively. Finally, chapter6considers the asymptotic behavior of stochastic retarded latticedynamical systems. We first present some sufficient conditions for theexistence of a global random attractor for random dynamical systemsdefined on the separable Banach space C ([ν,0],lpρ). Then we applythe obtained abstract results to the first-order stochastic retarded latticesystem with random coupled coefficients and additive white noises,and prove that this system possesses a random attractor.
The third part is chapter7, this chapter is to summarize the mainresults obtained in this paper, and propose some problems for futureresearch.
动力系统理论中最基本的、最重要的研究课题之一就是研究动力系统的渐近行为.随机吸引子是描述随机动力系统的渐近行为的有效工具.本文主要研究由带有Brown运动的几类随机微分方程生成的随机动力系统以及随机时滞格点动力系统的渐近行为,考虑系统的随机吸引子的存在性问题.这不仅具有重要的理论意义,而且具有重要的应用意义.
全文分为三个部分:
第一部分是本文的前两章,首先,第一章阐述本文的研究背景和现状以及本文的研究内容和意义.然后,第二章介绍与本文相关的一些基础知识.
第二部分,是本论文的核心内容,包括第三、四、五、六章.首先,第三章研究具有齐次Neumann边界条件的随机强阻尼波动方程的渐近行为,研究由此方程生成的随机动力系统的随机吸引子的存在性.其次,第四、五章分别研究定义在无界区域上的带有可乘噪声的随机阻尼波动方程和随机反应扩散方程的渐近行为,考虑由方程生成的随机动力系统的随机吸引子的存在性.最后,第六章研究随机时滞格点动力系统的渐近行为,首先给出在可分Banach空间C ([ν,0], lρp)上随机动力系统存在随机吸引子的充分条件,然后把所得的抽象结果应用到带有随机耦合系数和可加噪声的一阶随机时滞格点系统,证明此系统存在随机吸引子.
第三部分,即本文的第七章,本章对全文进行总结,并提出今后有待进一步研究的一些问题.
Dynamical system is the mathematical science which study on theevolution of systems. Inreality, many systems will inevitably be affectedby random factors. Considering the random factors in the dynamicalsystem will produce random dynamical system. Random dynamicalsystems appear in many important practical applications, such asphysics, mechanics, oceanography, meteorology, biology,communications engineering, other science and engineeringtechnology.
One of the most basic and important research topics of dynamicalsystems is to study the asymptotic behavior of dynamical systems.Random attractor is an effective tool to describe the asymptoticbehavior of random dynamical systems. This paper is devoted to studythe asymptotic behavior of random dynamical systems generated byseveral classes of stochastic differential equations driven by Brownmovement, as well as the asymptotic behavior of stochastic retardedlattice dynamical systems, consider the existence of random attractorsof these systems. This is important not only in theory, but also inapplication.
This paper is divided into three parts:
The first part is the first two chapters. First of all, chapter1introducesthe background and status of this study, as well as the content andsignificance of this study. Then, chapter2describes some basicknowledge related to this paper. The second part is the core content ofthis paper, including the chapters3,4,5and6. Firstly, chapter3isdevoted to study the asymptotic behavior of the stochastic stronglydamped wave equation with homogeneous Neumann boundarycondition. We investigate the existence of a random attractor for therandom dynamical system associated with the equation. Secondly, chapter4and chapter5study the asymptotic behavior of stochasticdamped wave equation and the stochastic reaction–diffusionequation with multiplicative noise on unbounded domain, and considerthe existence of random attractors for the random dynamical systemsgenerated by the two equations, respectively. Finally, chapter6considers the asymptotic behavior of stochastic retarded latticedynamical systems. We first present some sufficient conditions for theexistence of a global random attractor for random dynamical systemsdefined on the separable Banach space C ([ν,0],lpρ). Then we applythe obtained abstract results to the first-order stochastic retarded latticesystem with random coupled coefficients and additive white noises,and prove that this system possesses a random attractor.
The third part is chapter7, this chapter is to summarize the mainresults obtained in this paper, and propose some problems for futureresearch.
引文
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[2] R. Adams, J. Fournier, Sobolev Spaces, second ed., Elsevier Ltd.,2003.
[3] A. Babin, B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an un-bounded domain, J. Dyn. Diff. Eq.7(1995)567-590.
[4] J.M. Ball, Global attractors for damped semilinear wave equations, Disc. Contin. Dyn. Syst.10(2004)31-52.
[5] P.W. Bates, K. Lu, B. Wang, Random attractors for stochastic reaction-diffusion equationson unbounded domains, J. Diff. Eq.246(2009)845-869.
[6] P.W. Bates, K. Lu, B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur.Chaos11(2001)143-153.
[7] P.W. Bates, H. Lisei, K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn.6(2006)1-21.
[8] Z. Brzezniak, Y. Li, Asymptotic compactness and absorbing sets for2D Stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc.358(2006)5587-5629.
[9] T. Canaballo, J.A.Langa, Stability and random attractors for a reaction-diffusion equationwith multiplieative noise, Disc. Contin. Dyn. Syst.6(2000),875-892.
[10] T. Caraballo, J.A. Langa, J.C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, Proc. R. Soc. Lond. Ser. A457(2001)2041-2061.
[11] T. Caraballo, K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicativenoise, Front. Math. China3(2008)317-335.
[12] T. Caraballo, J. Real, Navier-Stokes equations with delays, Roy. Soc. London Proc. Ser. A.Math. Phys. Eng. Sci.457(2001)2441-2453.
[13] T. Caraballo, J. Real, Asymptotic behavior of Navier-Stokes equations with delays, Roy.Soc. London Proc. Ser. A. Math. Phys. Eng. Sci.459(2003)3181-3194.
[14] T. Caraballo, J. Real, Attractors for2D-Navier-Stokes models with delays, J. Diff. Eq.205(2004)271-297.
[15] T. Caraballo, P. Mar′n-Rubio, J. Valero, Autonomous and non-autonomous attractors fordifferential equations with delays, J. Diff. Eq.208(2005)9-41.
[16] V. Chepyzhov, M. Vishik, Trajectory attractors for reaction–diffusion systems, Topol. Meth-ods Nonlinear Anal.7(1996)49-76.
[17] F. Chen, B. Guo, P. Wang, Long time behavior of strongly damped nonlinear wave equations,J. Diff. Eq.147(1998)339-352.
[18] P. Chow, Stochastic wave equation with polynomial nonlinearity, Ann. Appl. Probab.12(2002)361-381.
[19] S. Chow, J. Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEETrans. Circuits Syst.42(1995)746-751.
[20] S. Chow, J. Paret, W. Shen, Traveling waves in lattice dynamical systems, J. Diff. Eq.149(1998)248-291.
[21] I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, NewYork,2002.
[22] H. Crauel, Random Probability Measure on Polish Spaces, Taylor&Francis, London,2002.
[23] H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dyn. Diff. Eq.9(1997)307-341.
[24] H. Crauel, F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields,100(1994)365-393.
[25] A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl.15(1997)473-491.
[26]丁孝全,随机/非自治泛函微分方程的渐近形态,同济大学博士学位论文,2009.
[27] J. Duan, K. Lu, B. Schmalfuss, Invariant manifolds for stochastic partial differential equa-tions, Ann. Probab.31(2003)2109-2135.
[28] X. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J.Math.216(2004)63-76.
[29] X. Fan, Y. Wang, Fractal dimensional of attractors for a stochastic wave equation with non-linear damping and white noise, Stoch. Anal. Appl.25(2007)381-396.
[30] X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise,Int. J. Math.19(2008)421-437.
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