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 and Continuous Dynamical Systems, 2019, 39 (7) : 4091-4126. doi: 10.3934/dcds.2019165
References:
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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.

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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.

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L. Caffarelli, J. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., (JEMS) 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.

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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.

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

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

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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.

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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. 

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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.

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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.

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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.

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

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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.

[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. 

[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.

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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.

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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. 

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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.

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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.

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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.

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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. 

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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.

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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.

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

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G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev., 36 (1930), 823. doi: 10.1103/PhysRev.36.823.

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B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

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B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$, Tran. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

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B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

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B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.  doi: 10.1016/j.na.2017.04.006.

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B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. 

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L. WangY. Wang and Y. Qin, Upper semi-continuity of attractors for nonclassical diffusion equations in $H(\mathbb{R}^3)$, Appl. Math. Comput., 240 (2014), 51-61.  doi: 10.1016/j.amc.2014.04.092.

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Z. WangS. Zhou and A. Gu, Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Anal. RWA, 12 (2011), 3468-3482.  doi: 10.1016/j.nonrwa.2011.06.008.

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X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

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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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[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.

[11]

L. Caffarelli, J. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., (JEMS) 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.

[12]

T. Caraballo and J. A. Langa, Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Disrete Contin. Dyn. Syst., 6 (2000), 875-892.  doi: 10.3934/dcds.2000.6.875.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[26]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[45]

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

[46]

G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev., 36 (1930), 823. doi: 10.1103/PhysRev.36.823.

[47]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[48]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$, Tran. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[49]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[50]

B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.  doi: 10.1016/j.na.2017.04.006.

[51]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. 

[52]

L. WangY. Wang and Y. Qin, Upper semi-continuity of attractors for nonclassical diffusion equations in $H(\mathbb{R}^3)$, Appl. Math. Comput., 240 (2014), 51-61.  doi: 10.1016/j.amc.2014.04.092.

[53]

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