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On the oscillation behavior of solutions to the one-dimensional heat equation
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 |
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.
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.
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[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. |
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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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Y. Wang, Z. Zhu and P. Li,
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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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Y. Xie, Q. Li and K. Zhu,
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M. Yang and P. E. Kloeden,
Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Anal. RWA, 12 (2011), 2811-2821.
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M. Yang, J. Duan and P. E. Kloeden,
Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. RWA, 12 (2011), 464-478.
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S. Zhou,
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S. Zhou and M. Zhao,
Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2017), 2887-2914.
|
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.
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