American Institute of Mathematical Sciences

Advanced
February  2019, 39(2): 683-706. doi: 10.3934/dcds.2019028

Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains

Hong Lu 1, , Jiangang Qi 1, , Bixiang Wang 2, and  Mingji Zhang 2,

1. 

School of Mathematics and Statistics, Shandong University, Weihai, Shandong Provence, 264209, China
2. 

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

Received  April 2017 Revised  January 2018 Published  November 2018

We study the asymptotic behavior of a class of non-autonomous non-local fractional stochastic parabolic equation driven by multiplicative white noise on the entire space $\mathbb{R}^n$. We first prove the pathwise well-posedness of the equation and define a continuous non-autonomous cocycle in $L^2({\mathbb{R}} ^n)$. We then prove the existence and uniqueness of tempered pullback attractors for the cocycle under certain dissipative conditions. The periodicity of the tempered attractors is also proved when the deterministic non-autonomous external terms are periodic in time. The pullback asymptotic compactness of the cocycle in $L^2({\mathbb{R}} ^n)$ is established by the uniform estimates on the tails of solutions for sufficiently large space and time variables.

Keywords: Fractional stochastic equation, random attractor, pullback asymptotic compactness.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.
Citation: Hong Lu, Jiangang Qi, Bixiang Wang, Mingji Zhang. Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 683-706. doi: 10.3934/dcds.2019028
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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.  Google Scholar

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J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst. Ser. A, 24 (2009), 855-882.  doi: 10.3934/dcds.2009.24.855.  Google Scholar

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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.  Google Scholar

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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, Comm. Math. Sci., 14 (2016), 273-295.  doi: 10.4310/CMS.2016.v14.n1.a11.  Google Scholar

[41]

H. Lu, S. Lv and M. Zhang, Fourier spectral approximation to the dynamical behavior of 3D fractional Ginzburg-Landau equation, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 2539-2564.  doi: 10.3934/dcds.2017109.  Google Scholar

[42]

H. Lu and M. Zhang, The spectral method for long-time behavior of a fractional power dissipative system, Taiwanese J. Math., 22 (2018), 453-483.  doi: 10.11650/tjm/170902.  Google Scholar

[43]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.  Google Scholar

[44]

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

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B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185–192, Dresden, 1992. Google Scholar

[46]

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

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R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

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B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

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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.  Google Scholar

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

[52]

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.  Google Scholar

[53]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.  Google Scholar

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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.  Google Scholar

show all references

References:
[1]

S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407.   Google Scholar

[2]

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

[3]

L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[4]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[5]

P. W. Bates, K. Lu and B. 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

[6]

W. J. Beyn, B. Gess, P. Lescot and M. R$\ddot o$ckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.  Google Scholar

[7]

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

[8]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

[9]

T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

[10]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Ser. A: Mathematical Analysis, 10 (2003), 491-513.   Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[14]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst., 19 (2004), 127-144.  doi: 10.1080/1468936042000207792.  Google Scholar

[15]

I. Chueshow, Monotone Random Systems - Theory and Applications, Lecture Notes in Mathematics, 1779, Springer, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[16]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[17]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[18]

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.  Google Scholar

[19]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.   Google Scholar

[20]

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

[21]

C. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.  Google Scholar

[22]

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

[23]

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.  Google Scholar

[24]

M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, Int. J. Bifur. Chaos, 20 (2010), 2761-2782.  doi: 10.1142/S0218127410027349.  Google Scholar

[25]

A. Garroni and S. Muller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal., 181 (2006), 535-578.  doi: 10.1007/s00205-006-0432-7.  Google Scholar

[26]

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.  Google Scholar

[27]

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

[28]

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

[29]

Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9.  Google Scholar

[30]

Q. Guan and Z. Ma, Reflected symmetric $α$-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694.  doi: 10.1007/s00440-005-0438-3.  Google Scholar

[31]

Q. Guan and Z. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.  doi: 10.1142/S021949370500150X.  Google Scholar

[32]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst. Ser. A, 24 (2009), 855-882.  doi: 10.3934/dcds.2009.24.855.  Google Scholar

[33]

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

[34]

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

[35]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176, Amer. Math. Soc., Providence, 2011. doi: 10.1090/surv/176.  Google Scholar

[36]

M. Koslowski, A. Cuitino and M. Ortiz, A phasefield theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystal, J. Mech. Phys. Solids, 50 (2002), 2597-2635.  doi: 10.1016/S0022-5096(02)00037-6.  Google Scholar

[37]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.  Google Scholar

[38]

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.   Google Scholar

[39]

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.  Google Scholar

[40]

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, Comm. Math. Sci., 14 (2016), 273-295.  doi: 10.4310/CMS.2016.v14.n1.a11.  Google Scholar

[41]

H. Lu, S. Lv and M. Zhang, Fourier spectral approximation to the dynamical behavior of 3D fractional Ginzburg-Landau equation, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 2539-2564.  doi: 10.3934/dcds.2017109.  Google Scholar

[42]

H. Lu and M. Zhang, The spectral method for long-time behavior of a fractional power dissipative system, Taiwanese J. Math., 22 (2018), 453-483.  doi: 10.11650/tjm/170902.  Google Scholar

[43]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.  Google Scholar

[44]

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

[45]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185–192, Dresden, 1992. Google Scholar

[46]

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

[47]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

[48]

Z. Shen, S. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.  Google Scholar

[49]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[50]

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.  Google Scholar

[51]

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

[52]

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