July  2019, 39(7): 4091-4126. doi: 10.3934/dcds.2019165

Random dynamics of fractional nonclassical diffusion equations driven by colored noise

1. 

School of Mathematics and statistics, Southwest University, Chongqing 400715, China

2. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: liyr@swu.edu.cn (Yangrong Li)

Received  October 2018 Revised  January 2019 Published  April 2019

The random dynamics in $ H^s(\mathbb{R}^n) $ with $ s\in (0,1) $ is investigated for the fractional nonclassical diffusion equations driven by colored noise. Both existence and uniqueness of pullback random attractors are established for the equations with a wide class of nonlinear diffusion terms. In the case of additive noise, the upper semi-continuity of these attractors is proved as the correlation time of the colored noise approaches zero. The methods of uniform tail-estimate and spectral decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embedding on an unbounded domain.

Citation: Renhai Wang, Yangrong Li, Bixiang Wang. Random dynamics of fractional nonclassical diffusion equations driven by colored noise. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4091-4126. doi: 10.3934/dcds.2019165
References:
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show all references

References:
[1]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.  doi: 10.3934/dcdsb.2013.18.643.  Google Scholar

[2]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mechanica, 37 (1980), 265-296.  doi: 10.1007/BF01202949.  Google Scholar

[3]

E. C. Aifantis adient nanomechanics, applications to deformation, fracture, and diffusion in nanopolycrystals, Metallurgical and Materials Transactions A, 42 (2011), 2985-2998.   Google Scholar

[4]

C. T. Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, Nonlinear Anal., 73 (2010), 399-412.  doi: 10.1016/j.na.2010.03.031.  Google Scholar

[5]

C. T. Anh and T. Q. Bao, Dynamics of non-autonomous nonclassical diffusion equations on $\mathbb{R}^n$, Commun. Pure Appl. Anal., 11 (2012), 1231-1252.  doi: 10.3934/cpaa.2012.11.1231.  Google Scholar

[6]

V. Anishchenko, V. Astakhov, A. Neiman, T. Vadivasova and L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments, Springer Series in Synergetics. Springer-Verlag, Berlin, 2002.  Google Scholar

[7]

L. Bai and F. Zhang, Existence of random attractors for 2D-stochastic nonclassical diffusion equations on unbounded domains, Results Math., 69 (2016), 129-160.  doi: 10.1007/s00025-015-0505-8.  Google Scholar

[8]

P. W. BatesK. Lu and B. X. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[9]

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[10]

P. W. Bates, K. Lu and B. Wang, Tempered random attractors for parabolic equations in weighted spaces, J. Math. Phys., 54 (2013), 081505, 26 pp. doi: 10.1063/1.4817597.  Google Scholar

[11]

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[12]

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[13]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[14]

T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.  Google Scholar

[15]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.  Google Scholar

[16]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[17]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[18]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[19]

H. Gao and C. Sun, Random dynamics of the 3D stochastic Navier-Stokes-Voight equations, Nonlinear Anal. RWA, 13 (2012), 1197-1205.  doi: 10.1016/j.nonrwa.2011.09.013.  Google Scholar

[20]

M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differ. Equ., 23 (2011), 671-681.  doi: 10.1007/s10884-011-9222-5.  Google Scholar

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M. J. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.  Google Scholar

[22]

W. Gerstner, W. Kistler, R. Naud and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition, Cambridge University Press, Cambridge, 2014. Google Scholar

[23]

B. GessW. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.  Google Scholar

[24]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dynam. Differ. Equ., 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.  Google Scholar

[25]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.  Google Scholar

[26]

A. Gu, B. Guo and B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, (2018), submitted. Google Scholar

[27]

A. GuD. LiB. Wanga and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.  Google Scholar

[28]

A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.   Google Scholar

[29]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.  doi: 10.1002/cpa.20253.  Google Scholar

[30]

R. Jones and B. Wang, Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. RWA, 14 (2013), 1308-1322.  doi: 10.1016/j.nonrwa.2012.09.019.  Google Scholar

[31]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[32]

K. Kuttler and E. C. Aifantis, Quasilinear evolution equations in nonclassical diffusion, SIAM J. Math. Anal., 19 (1998), 110-120.  doi: 10.1137/0519008.  Google Scholar

[33]

D. LiB. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.  Google Scholar

[34]

D. LiK. LuB. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.   Google Scholar

[35]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[36]

Y. Li and R. Wang, Random attractors for 3D Benjamin-Bona-Mahony equations derived by a Laplace-multiplier noise, Stoch. Dyn., 18 (2018), 1850004, 26pp. doi: 10.1142/S0219493718500041.  Google Scholar

[37]

Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.   Google Scholar

[38]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differ. Equ., (2017), 1-31. doi: 10.1007/s10884-017-9626-y.  Google Scholar

[39]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of 3D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.  Google Scholar

[40]

H. LuP. W. BatesJ. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on $\mathbb{R}^n$, Nonlinear Anal., 128 (2015), 176-198.  doi: 10.1016/j.na.2015.06.033.  Google Scholar

[41]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.   Google Scholar

[42]

C. Morosi and L. Pizzocchero, On the constants for some fractional Gagliardo-Nirenberg and Sobolev inequalities, Expo. Math., 36 (2018), 32-77.  doi: 10.1016/j.exmath.2017.08.007.  Google Scholar

[43]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[44]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783.  Google Scholar

[45]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[46]

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