Random dynamics of fractional nonclassical diffusion equations driven by colored noise

Renhai Wang; Yangrong Li; Bixiang Wang

doi:10.3934/dcds.2019165

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

Random dynamics of fractional nonclassical diffusion equations driven by colored noise

Renhai Wang ^1, , Yangrong Li ^1,, and Bixiang Wang ^2,

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.

Keywords: Nonclassical diffusion equation, fractional Laplace operator, colored noise, random attractor, upper semi-continuity.

Mathematics Subject Classification: Primary: 37L55; Secondary: 35B40, 60H15.

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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[34]	D. Li, K. Lu, B. 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. Li, A. 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.
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[40]	H. Lu, P. W. Bates, J. 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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[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.
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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. Bates, K. 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. Bates, K. 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. Caraballo, M. J. Garrido-Atienza, B. 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. Caraballo, M. 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. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225.
[17]	E. Di Nezza, G. 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-Atienza, A. 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. Gess, W. 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. Gu, D. Li, B. 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. Li, B. 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. Li, K. Lu, B. 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. Li, A. 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. Lu, P. W. Bates, S. 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. Lu, P. W. Bates, J. 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. Lu, P. W. Bates, S. 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. Wang, Y. 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]	Z. Wang, S. 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.
[54]	Y. Wang, Z. Zhu and P. Li, Regularity of pullback attractors for nonautonomous nonclassical diffusion equations, J. Math. Anal. Appl., 459 (2018), 16-31. doi: 10.1016/j.jmaa.2017.10.075.
[55]	X. Wang, K. 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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Random dynamics of fractional nonclassical diffusion equations driven by colored noise

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Random dynamics of fractional nonclassical diffusion equations driven by colored noise

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