无界区域非自治随机sine-Gordon方程的D-周期吸引子
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摘要
研究非自治随机sine-Gordon方程所生成的随机动力系统φ的D-周期吸引子的存在唯一性.运用一致估计得到了D的D-吸收集的存在性,并用分解技巧,证明了φ的渐近紧性,建立了动力系统φ在H~1(R~n)×L~2(R~n)中的D-周期吸引子的存在唯一性.当非自治外力项具有周期性时,该D-吸引子也呈现相同的周期性.
We concerns on the existence and uniqueness of periodic-attractor for random dynamical system φ generated by sine-Gordon equation with non-autonomous term as well as stochastic term.Firstly,a D-pullback absorbing set of φ is obtained by uniform estimates.Secondly,by applying a splitting technique,the asymptotic compactness of φ is proved.Finally,the existence and uniqueness of periodic D-attractor of the corresponding random dynamical system in H~1(R~n) × L~2{R~n) is established.Moreover,the D-pullback attractor is proved to be periodic when the random dynamical system contains a periodic deterministic forcing term.
We concerns on the existence and uniqueness of periodic-attractor for random dynamical system φ generated by sine-Gordon equation with non-autonomous term as well as stochastic term.Firstly,a D-pullback absorbing set of φ is obtained by uniform estimates.Secondly,by applying a splitting technique,the asymptotic compactness of φ is proved.Finally,the existence and uniqueness of periodic D-attractor of the corresponding random dynamical system in H~1(R~n) × L~2{R~n) is established.Moreover,the D-pullback attractor is proved to be periodic when the random dynamical system contains a periodic deterministic forcing term.
引文
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[2]Ball J.M.,Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations,J.Nonlinear Sci.,1997,7:475-502.
[3]Ball J.M.,Global attractors for damped semilinear wave equation,Discrete Continuous Dynamical Systems,2004,10:31-52.
[4]Caraballo T.,Lukaszewicz G.,Real J.,Pullback attractors for asymptotically compact non-autonomous dynamical systems,Nonlinear Anal.,2006,64:484-498.
[5]Caraballo T.,Lukaszewicz G.,Real J.,Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains,C.R.Math.Acad.Sci.Paris I,2006,342:263-268.
[6]Caraballo T.,Real J.,Chueshov I.D.,Pullback attractors for stochastic heat equations in materials with memory,Discrete Contin.Dynam.Syst.B,2008,9:525-539.
[7]Cheban D.N.,Global Attractors of Non-autonomous Dissipative Dynamical Systems,World Scientific,Singapore,2004.
[8]Chepyzhov V.V.,Vishik M.I.,Attractors of non-autonomous dynamical systems and their dimensions,J.Math.Pures.Appl.,1994,73:279-333.
[9]Chow P.L.,Asymptotic of solutions to semilinear stochastic wave equations,Analysis and Applied Probability,2006,2:757-789.
[10]Crauel H.,Debussche A.,Flandoli F.,Random attractors,J.Dyanm.Differential Equations,1997,9:307-341.
[11]Crauel H.,Flandoli F.,Attractors for random dynamical systems,Probab.Th.Re.Fields,1994,100:365-393.
[12]Fan X.,Random attractor for a damped sine-Gordon equation with white noise,Pacifc.J.Math.,2004,216:63-76.
[13]Feireisl E.,Attractors for semilinear damped wave equations on R~3,Nonlinear Analysis TMA,1994,23:187-195.
[14]Huang J.,Shen W.,Pullback attractors for non-autonomous and random paraboUc equations on non-smooth domains,Discrete Contin.Dynam.Syst.,2009,24:855-882.
[15]Marinelli C,Sardanyons L.Q.,Existence of weak solutions for a class of semilinear stochastic wave equations,SIAM J.Math.Anal.,2012,2:906-925.
[16]Rosa R.,The global attractor for the 2D Navier-Stokes flow on some unbounded domains,Nonlinear Analysis,TMA,1998,32:71-85.
[17]Song H.T.,Pullback attractors of non-autonomous reaction-diffusion equations in H_0~1,J.Differential Equations,2010,249:2357-2376.
[18]Temam R.,Infinite Dimensional Dynamical Systems in Mechanics and Physics,Spring-Verlag,New York,1998.
[19]Wang B.X.,Asymptotic behavior of stochastic wave equations with critical exponents on R~3,Trans.Amer.Math.Soc,2011,36:3639-3663.
[20]Wang B.X.,Periodic random attractors for stochastic Navier-Stokes equation on unbounded domains,J.Differential Equations,2012,59:1-18.
[21]Wang B.X.,Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains,Nonlinear Analysis,2009,70:3799-3815.
[22]Wang B.X.,Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical system,J.Differential Equations,2012,253:1544-1583.
[23]Wang B.X.,Gao X.L.,Random Attractors for wave Equations on Unbounded Domains,Discrete and Continuous Dynamical Systems,September 2009,Issue Special:800-809.
[24]Wang Z.J.,Zhou S.F.,Gu A.H.,Random attractor of the stochastic strongly damped wave equation,Communications in Nonlinear Science and Numerical Simulation,2012,17:1649-1658.
[25]Zhou S.F.,Attractors for strongly damped wave equations with critical exponent,Applied Mathematics Letters,2003,16:1307-1314.
[26]Zhou S.F.,Shen W.,One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation,J.Differential Equations,2010,248:1432-1457.