无界区域上具可乘白噪音的Fitzhngh-Nagumo方程的渐近行为
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摘要
主要研究了定义在无界区域上具可乘白噪音的Fitzhugh-Nagumo方程的渐近行为.首先运用Ornstein-Uhlenbeck变换,将Fitzhugh-Nagumo方程转换成带有随机参数的确定型系统,并生成了相应的随机动力系统.其次运用一致估计证明了所生成的随机动力系统的渐近性.最后,证明了该随机动力系统的随机吸引子的存在性.
This paper is devoted to investigating the asymptotic behavior for Fitzhugh-Nagumo systems with multiplicative white noise on unbounded domains.Firstly,by using Ornstein-Uhlenbeck transformation,the Fitzhugh-Nagumo systems are transferred into a deterministic case with random parameter and generate the corresponding random dynamical system.Secondly,applying the uniform priori estimates to solutions,we prove the asymptotic compactness of the mentioned random dynamical system.Finally,the existence of a random attractor for this random dynamical system is established.
This paper is devoted to investigating the asymptotic behavior for Fitzhugh-Nagumo systems with multiplicative white noise on unbounded domains.Firstly,by using Ornstein-Uhlenbeck transformation,the Fitzhugh-Nagumo systems are transferred into a deterministic case with random parameter and generate the corresponding random dynamical system.Secondly,applying the uniform priori estimates to solutions,we prove the asymptotic compactness of the mentioned random dynamical system.Finally,the existence of a random attractor for this random dynamical system is established.
引文
[1]WANG B.Random attractors for the stochastic Fitzhugh-Nagumo system on unbounded domains[J].Nonlinear Anal,2009,8:2 811-2 828.
[2]WANG Z,ZHOU S.Random attractors for the stochastic reaction-diffusion equation with multiplicative noise on unbounded domains[J].J Math Anal Appl,2011,384:160-172.
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[6]BABIN A V,VISHISK M I.Attractors of Evolution Equations[M].Amsterdam:Norht-Holland,1992.
[7]ADAMS R.Sobolev Spaces[M].New York:Academic Press,1975.
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[9]BATES P W,LU K,WANG B.Random attractors for stochastic reaction-diffusion equations on unbounded domains[J].J Differential Equations,2009.
[10]CARABALLO T,LANGA J A,ROBISION J C.A stochastic pitchfork bifurcation in a reaction-diffusion equations[J].Proc R Soc Lond A,2001,457:2 041-2 061.
[11]HALE J K.Asymptotic behavior of dissipative systems[M].Providence,RI:American Mathematical Society,1988.
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[14]TEMAM R.Infinite dimensional dynamical systems in mechanics and physics[M].New York:Spring-Verlag,1998.
[2]WANG Z,ZHOU S.Random attractors for the stochastic reaction-diffusion equation with multiplicative noise on unbounded domains[J].J Math Anal Appl,2011,384:160-172.
[3]ARNOLD L.Random Dynamical Systems[M].NewYork:Springer-Verlag,1998.
[4]BALL J M.Continuity properties and global attractors of gernerlized semiflows and the Naiver-Stokes equations[J].J Nonlinear Sci,1997,7:475-502.
[5]BALL J M.Global attractors for damped semilinear wave equations[J].Discrete Contin Dyn Syst,2004,10:31-52.
[6]BABIN A V,VISHISK M I.Attractors of Evolution Equations[M].Amsterdam:Norht-Holland,1992.
[7]ADAMS R.Sobolev Spaces[M].New York:Academic Press,1975.
[8]BATES P W,LISEI H,LU K.Attractors for the stochastic lattice dynamical systems[J].Stoch Dyn,2006(6):1-21.
[9]BATES P W,LU K,WANG B.Random attractors for stochastic reaction-diffusion equations on unbounded domains[J].J Differential Equations,2009.
[10]CARABALLO T,LANGA J A,ROBISION J C.A stochastic pitchfork bifurcation in a reaction-diffusion equations[J].Proc R Soc Lond A,2001,457:2 041-2 061.
[11]HALE J K.Asymptotic behavior of dissipative systems[M].Providence,RI:American Mathematical Society,1988.
[12]CRAUEL H,FLANDOLI F.Attractors for random dynamical systems[J].Probab Th Re Fields,1994,100:365-393.
[13]FLANDOLI F,SCHMALFUBβB.Random attractors for the 3Dstochastic Navier-Stokes equation with multiplicative noise[J].Stoch Rep,1996,59:21-45.
[14]TEMAM R.Infinite dimensional dynamical systems in mechanics and physics[M].New York:Spring-Verlag,1998.