三分量可逆Gray-Scott系统及随机格点动力系统的吸引子
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摘要
吸引子是一个用以描述无穷维动力系统解的长时间渐近行为的最恰当工具.三分量可逆Gray-Scott系统是很重要的一类反应扩散方程,它用以描述两个可逆化学或生化反应.本文主要研究三分量可逆Gray-Scott系统吸引子的存在性问题及带有分数Brown噪声、Levy噪声的随机格点动力系统的随机吸引子的存在性.这些问题的研究可为相关领域如化学、生化反应、生物、材料科学等提供新的视角,丰富了动力系统的研究范畴.
全文分为三个部分:
第一部分由前两章构成.
第一章给出本文的研究背景、现状及本文的研究内容和意义,
第二章介绍一些相关的准备知识.
第二部分主要研究三分量可逆Gray-Scott系统的吸引子的存在性问题.
第三章给出具有Dirichlet边界条件的非自治三分量可逆Gray-Scott系统的一致吸引子的存在性及无界区域上拉回吸引子的存在性.
第四章分别建立随机三分量可逆Gray-Scott系统在有界、无界区域上随机吸引子的存在性.
第三部分由第五章组成,分别研究了随机三分量可逆Gray-Scott格点系统及一阶随机格点动力系统在分数Brown噪声及Levy噪声扰动下随机吸引子的存在性问题.
It is well known that attractor is the appropriate tool to study the long timeasymptotic behaviors of the solutions of infinite dimensional dynamical systems.Among the reaction-diffusion systems, the three-component reversible Gray-Scottsystem is a vita model to describe the scheme of two reversible chemical orbiochemical reactions. In this doctoral dissertation, we mainly consider the existenceof global attractors for the three-component reversible Gray-Scott system and thelattice dynamical systems when perturbed by the fractional Brownian noise and Levynoise. These results not only can enrich the contents in the research of the dynamicalsystems, but also can provide new insights to the related research areas, such as,chemistry, biochemistry, biology, science of material, etc.
The first section consists of the first two chapters. In chapter1, we firstintroduce the background and status of these aspects of research, as well as thecontents and significance of this study. Chapter2recalls some preliminary results.
In section2, we mainly discuss the existence of global attractors for thethree-component reversible Gray-Scott system. Chapter3considers the uniformattractor for the non-autonomous three-component reversible Gray-Scott systemwith Dirichlet boundary condition on a bounded domain and the pullback attractoron unbounded domains. Chapter4obtains the existence of random attractors for thestochastic three-component reversible Gray-Scott system on bounded domain andunbounded domains, respectively.
Section3comprises chapters5. We respectively establish the existence of globalrandom attractors of the stochastic three-component reversible Gray-Scott system,the first-order stochastic lattice dynamical system with fractional Brownian noise and Levy noise.
全文分为三个部分:
第一部分由前两章构成.
第一章给出本文的研究背景、现状及本文的研究内容和意义,
第二章介绍一些相关的准备知识.
第二部分主要研究三分量可逆Gray-Scott系统的吸引子的存在性问题.
第三章给出具有Dirichlet边界条件的非自治三分量可逆Gray-Scott系统的一致吸引子的存在性及无界区域上拉回吸引子的存在性.
第四章分别建立随机三分量可逆Gray-Scott系统在有界、无界区域上随机吸引子的存在性.
第三部分由第五章组成,分别研究了随机三分量可逆Gray-Scott格点系统及一阶随机格点动力系统在分数Brown噪声及Levy噪声扰动下随机吸引子的存在性问题.
It is well known that attractor is the appropriate tool to study the long timeasymptotic behaviors of the solutions of infinite dimensional dynamical systems.Among the reaction-diffusion systems, the three-component reversible Gray-Scottsystem is a vita model to describe the scheme of two reversible chemical orbiochemical reactions. In this doctoral dissertation, we mainly consider the existenceof global attractors for the three-component reversible Gray-Scott system and thelattice dynamical systems when perturbed by the fractional Brownian noise and Levynoise. These results not only can enrich the contents in the research of the dynamicalsystems, but also can provide new insights to the related research areas, such as,chemistry, biochemistry, biology, science of material, etc.
The first section consists of the first two chapters. In chapter1, we firstintroduce the background and status of these aspects of research, as well as thecontents and significance of this study. Chapter2recalls some preliminary results.
In section2, we mainly discuss the existence of global attractors for thethree-component reversible Gray-Scott system. Chapter3considers the uniformattractor for the non-autonomous three-component reversible Gray-Scott systemwith Dirichlet boundary condition on a bounded domain and the pullback attractoron unbounded domains. Chapter4obtains the existence of random attractors for thestochastic three-component reversible Gray-Scott system on bounded domain andunbounded domains, respectively.
Section3comprises chapters5. We respectively establish the existence of globalrandom attractors of the stochastic three-component reversible Gray-Scott system,the first-order stochastic lattice dynamical system with fractional Brownian noise and Levy noise.
引文
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[4]黄建华,黎育红,郑言,随机动力系统引论,科学出版社,2011.
[5] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York,1998.
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[21] P.W. Bates, H. Lisei, K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn.6(2006)1-21.
[22] 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.
[23] T. Caraballo, K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicativenoise, Front. Math. China3(2008)317-335.
[24] T. Caraballo, G. ukaszewicz, J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal.64(2006)484-498.
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[28] H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dyn. Diff. Equat.9(1997)307-341.
[29] H. Crauel, F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields,100(1994)365-393.
[30] M. Garrido-Atienza, P. Kloeden, A. Neuenkirch, Discretization of stationary solutions ofstochastic systems driven by a fractional Brownian motion, Appl. Math. Optim.60(2009)151–172.
[31] M. Garrido-Atienza, K. Lu, B. Schmalfu, Random dynamical systems for stochastic equa-tions driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst.14(2010)473–493.
[32] M. Garrido-Atienza, B. Maslowski, B. Schmalfu, Random attractors for stochastic equa-tions driven by a fractional Brownian motion, Internat. J. Bifur. Chaos Appl. Sci. Engrg.20(2010)2761–2782.
[33] B. Maslowski, B. Schmalfu, Random dynamical systems and stationary solutions of differ-ential equations driven by the fractional Brownian motion, Stochasic Anal. Appl.22(2004)1577–1607.
[34] X. Liu, J. Duan, J. Liu, P. Kloeden, Synchronization of systems of Marcus canonical equa-tions driven by α-stable noises, Nonlinear Anal. RWA.11(2010)3437–3445.
[35] F. Flandoli, H. Lisei, Stationary conjugation of flows for parabolic SPDEs with multiplicativenoise and some applications, Stochast. Anal. Appl.22(2004)1385–1420.
[36] J. Ball. Global attractors of damped semilinear wave equations, Discrete Contin. Dyn. Syst.10(2004)31-52.
[37] I. Moise, R. Rosa, B. Wang. Attractors for noncompact non-autonomous systems via energyequations, Discrete Contin. Dyn. Syst.10(2004)473-496.
[38] A. Rodriguez-Bernal, B. Wang. Attractors for partly dissipative reation diffusion systems inRn, J. Math. Anal. Appl.52(2000)790-803.
[39] M. Efendiev, S. Zelik, The attractor for a nonlinear reaction-diffusion system in an un-bounded domain, Comm. Pure Appl. Math.54(6)625-688(2001).
[40] S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatialcomplexity, Comm. Pure Appl. Math.56(5)584-637(2003).
[41] P. W. Bates, K. Lu, B. Wang, Random attractors for stochastic Reaction-Diffusion equationson unbounded domains, J. Diff. Eq.246(2009)845-869.
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