具强阻尼的随机波动方程随机吸引子的分形维数
|
摘要
主要考虑带可加与可乘白噪声的具强阻尼的随机波动方程的随机吸引子的分形维数的上界估计式.首先,利用Ornstein-Uhlenbeck过程将具白噪声的随机方程转化成以随机变量为参数的无噪声的随机方程;然后,把该随机方程的2个解之差适当分解成2个部分之和,并分别估计这2个部分的模及某些随机变量的期望的有界性;最后,得到了所研方程的随机吸引子的分形维数的上界估计式.
It was considered the upper bound estimation of the fractal dimension of random attractors for stochastic strongly damped wave equations with additive and multiplicative white noises. Firstly,these stochastic equations with white noises was transfered into random equations with random parameters without noise terms by the Ornstein-Uhlenbeck process. Secondly,it was divided the difference between the two solutions of random equations into a sum of two parts,and estimated the norm of these two parts and the boundedness of expectation of some random variables,respectively. Finally,it was obtained the upper bound estimation formula of the fractal dimension of random attractors for the considered equations.
It was considered the upper bound estimation of the fractal dimension of random attractors for stochastic strongly damped wave equations with additive and multiplicative white noises. Firstly,these stochastic equations with white noises was transfered into random equations with random parameters without noise terms by the Ornstein-Uhlenbeck process. Secondly,it was divided the difference between the two solutions of random equations into a sum of two parts,and estimated the norm of these two parts and the boundedness of expectation of some random variables,respectively. Finally,it was obtained the upper bound estimation formula of the fractal dimension of random attractors for the considered equations.
引文
[1]LüYan,Wang Wei.Limiting dynamics for stochastic wave equations[J].J Differential Equations,2008,244(1):1-23.
[2]Yang Meihua,Duan Jinqiao,Kloeden P.Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise[J].Nonlinear Anal Real World Appl,2011,12(1):464-478.
[3]Zhou Shengfan,Yin Fuqi,Ouyang Zigen.Random attractor for damped nonlinear wave equations with white noise[J].SIAM J Appl Dyn Syst,2005,4(4):883-903.
[4]Fan Xiaoming.Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise[J].Stoch Anal Appl,2006,24(4):767-793.
[5]Fan Xiaoming.Random attractors for damped stochastic wave equations with multiplicative noise[J].Internat J Math,2008,19(4):421-437.
[6]Fan Xiaoming,Wang Yaguang.Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise[J].Stoch Anal Appl,2007,25(2):381-396.
[7]Zhou Shengfan,Zhao Min.Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation[J].Nonlinear Anal,2016,133:292-318.
[8]Zhou Shengfan,Zhao Min.Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise[J].Discrete Contin Dyn Syst,2016,36(5):2887-2914.
[9]Ghidaglia J M,Marzocchi A.Longtime behavior of strongly damped nonlinear wave equations,global attractors and their dimension[J].SIAM J Math Anal,1991,22(3):879-895.
[10]Yang Meihua,Sun Chunyou.Dynamics of strongly damped wave equations in locally uniform spaces:Attractors and asymptotic regularity[J].Trans Amer Math Soc,2009,361(4):1069-1101.
[11]尹福其,周盛凡,李红艳.具强阻尼的随机sine-Gordon方程的随机吸引子存在性[J].上海大学学报:自然科学版,2006,12(3):260-265.
[12]郝红娟,周盛凡.强阻尼随机sine-Gordon方程随机吸引子的存在性[J].上海师范大学学报:自然科学版,2010,39(2):121-127.
[13]Wang Zhaojuan,Zhou Shengfan.Asymptotic behavior of stochastic strongly damped wave equation with multiplicative noise[J].International Journal of Modern Nonlinear Theory and Application,2015,4(3):204-214.
[14]Foias C,Olson E.Finite fractal dimension and H9lder-Lipschitz parametrization[J].Indiana Univ Math J,1996,45(3):603-616.
[2]Yang Meihua,Duan Jinqiao,Kloeden P.Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise[J].Nonlinear Anal Real World Appl,2011,12(1):464-478.
[3]Zhou Shengfan,Yin Fuqi,Ouyang Zigen.Random attractor for damped nonlinear wave equations with white noise[J].SIAM J Appl Dyn Syst,2005,4(4):883-903.
[4]Fan Xiaoming.Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise[J].Stoch Anal Appl,2006,24(4):767-793.
[5]Fan Xiaoming.Random attractors for damped stochastic wave equations with multiplicative noise[J].Internat J Math,2008,19(4):421-437.
[6]Fan Xiaoming,Wang Yaguang.Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise[J].Stoch Anal Appl,2007,25(2):381-396.
[7]Zhou Shengfan,Zhao Min.Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation[J].Nonlinear Anal,2016,133:292-318.
[8]Zhou Shengfan,Zhao Min.Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise[J].Discrete Contin Dyn Syst,2016,36(5):2887-2914.
[9]Ghidaglia J M,Marzocchi A.Longtime behavior of strongly damped nonlinear wave equations,global attractors and their dimension[J].SIAM J Math Anal,1991,22(3):879-895.
[10]Yang Meihua,Sun Chunyou.Dynamics of strongly damped wave equations in locally uniform spaces:Attractors and asymptotic regularity[J].Trans Amer Math Soc,2009,361(4):1069-1101.
[11]尹福其,周盛凡,李红艳.具强阻尼的随机sine-Gordon方程的随机吸引子存在性[J].上海大学学报:自然科学版,2006,12(3):260-265.
[12]郝红娟,周盛凡.强阻尼随机sine-Gordon方程随机吸引子的存在性[J].上海师范大学学报:自然科学版,2010,39(2):121-127.
[13]Wang Zhaojuan,Zhou Shengfan.Asymptotic behavior of stochastic strongly damped wave equation with multiplicative noise[J].International Journal of Modern Nonlinear Theory and Application,2015,4(3):204-214.
[14]Foias C,Olson E.Finite fractal dimension and H9lder-Lipschitz parametrization[J].Indiana Univ Math J,1996,45(3):603-616.