doi: 10.3934/dcdsb.2022081
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Upper semi-continuity of non-autonomous fractional stochastic $ p $-Laplacian equation driven by additive noise on $ \mathbb{R}^n $

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

*Corresponding author: Xuping Zhang

Received  February 2022 Early access April 2022

Fund Project: This work was supported by the Outstanding Youth Science Fund of Gansu Province (No. 21JR7RA159), the Natural Science Foundations of Gansu Province (20JR5RA522) and Project of NWNU-LKQN2019-13

This paper deals with the asymptotic behavior of the solutions to a class of non-autonomous fractional stochastic $ p $-Laplacian equation driven by linear additive noise on the entire space $ \mathbb{R}^n $. We firstly prove the existence of a continuous non-autonomous cocycle for the equation as well as the uniform estimates of solutions. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors and the uniform tail-estimates of the solutions for large space variables when time is large enough to surmount the lack of compact Sobolev embeddings on unbounded domains. Finally, we establish the upper semi-continuity of the random attractors when noise intensity approaches zero.

Citation: Xiaohui Zhang, Xuping Zhang. Upper semi-continuity of non-autonomous fractional stochastic $ p $-Laplacian equation driven by additive noise on $ \mathbb{R}^n $. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022081
References:
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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.

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P. W. BatesK. 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.

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P. Chen and X. Zhang, Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 4325-4357.  doi: 10.3934/dcdsb.2020290.

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P. Chen, X. Zhang and X. Zhang, Asymptotic behavior of non-autonomous fractional stochastic p-Laplacian equations with delay on $\mathbb{R}^n $, J. Dynam. Differential Equations, (2021). doi: 10.1007/s10884-021-10076-4.

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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.

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P. G. Geredeli, On the existence of regular global attractor for $p$-Laplacian evolution equation, Appl. Math. Optim., 71 (2015), 517-532.  doi: 10.1007/s00245-014-9268-y.

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B. GessW. Liu and M. Röckner, 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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A. GuD. LiB. Wang 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. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038.  doi: 10.1016/j.jmaa.2014.03.037.

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D. Li, B. Wang and X. Wang, Random dynamics of fractional stochastic reaction-diffusion equations on $\mathbb{R}^n $ without uniqueness, J. Math. Phys., 60 (2019), 21pp. doi: 10.1063/1.5063840.

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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.

[18]

Y. Li and Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.

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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. (9), 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[20]

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.

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J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[22]

M. Sui and Y. Wang, Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.  doi: 10.1016/j.amc.2014.05.045.

[23]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[24]

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.

[25]

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

[26]

B. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1-59.  doi: 10.1016/j.jde.2019.08.007.

[27]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 31pp. doi: 10.1142/S0219493714500099.

[28]

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.

[29]

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.

[30]

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.

[31]

R. Wang, Y. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019) 4091–4126. doi: 10.3934/dcds.2019165.

[32]

R. WangL. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R}^N $, Nonlinearity, 32 (2019), 4524-4556.  doi: 10.1088/1361-6544/ab32d7.

[33]

R. Wang and B. Wang, Asymptotic behavior of non-autonomous fractional p-Laplacian equations driven by additive noise on unbounded domains, Bull. Math. Sci., 11 (2021), 50pp. doi: 10.1142/S1664360720500204.

[34]

R. Wang and B. Wang, Asymptotic behavior of non-autonomous fractional stochastic p-Laplacian equations, Comput. Math. Appl., 78 (2019), 3527-3543.  doi: 10.1016/j.camwa.2019.05.024.

[35]

R. Wang and B. Wang, Random dynamics of non-autonomous fractional stochastic p-Laplacian equations on $\mathbb{R}^N $, Banach J. Math. Anal., 15 (2021), 42pp. doi: 10.1007/s43037-020-00107-5.

show all references

References:
[1]

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.

[2]

P. W. BatesK. 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.

[3]

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.

[4]

P. Chen, R. Wang and X. Zhang, Long-time dynamics of fractional nonclassical diffusion equations with nonlinear colored noise and delay on unbounded domains, Bull. Math. Sci., 173 (2021), 52pp. doi: 10.1016/j.bulsci.2021.103071.

[5]

P. Chen and X. Zhang, Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 4325-4357.  doi: 10.3934/dcdsb.2020290.

[6]

P. Chen, X. Zhang and X. Zhang, Asymptotic behavior of non-autonomous fractional stochastic p-Laplacian equations with delay on $\mathbb{R}^n $, J. Dynam. Differential Equations, (2021). doi: 10.1007/s10884-021-10076-4.

[7]

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.

[8]

P. G. Geredeli, On the existence of regular global attractor for $p$-Laplacian evolution equation, Appl. Math. Optim., 71 (2015), 517-532.  doi: 10.1007/s00245-014-9268-y.

[9]

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

[10]

B. GessW. Liu and M. Röckner, 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.

[11]

A. GuD. LiB. Wang 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.

[12]

J. HuangT. Shen and Y. Li, Dynamics of stochastic fractional Boussinesq equations, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2051-2067.  doi: 10.3934/dcdsb.2015.20.2051.

[13]

A. Kh. Khanmamedov, Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain, J. Math. Anal. Appl., 316 (2006), 601-615.  doi: 10.1016/j.jmaa.2005.05.003.

[14]

P. E. Kloeden, Upper semi continuity of attractors of delay differential equations in the delay, Bull. Austral. Math. Soc., 73 (2006), 299-306.  doi: 10.1017/S0004972700038880.

[15]

A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038.  doi: 10.1016/j.jmaa.2014.03.037.

[16]

D. Li, B. Wang and X. Wang, Random dynamics of fractional stochastic reaction-diffusion equations on $\mathbb{R}^n $ without uniqueness, J. Math. Phys., 60 (2019), 21pp. doi: 10.1063/1.5063840.

[17]

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.

[18]

Y. Li and Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.

[19]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[20]

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.

[21]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[22]

M. Sui and Y. Wang, Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.  doi: 10.1016/j.amc.2014.05.045.

[23]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[24]

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.

[25]

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

[26]

B. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1-59.  doi: 10.1016/j.jde.2019.08.007.

[27]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 31pp. doi: 10.1142/S0219493714500099.

[28]

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.

[29]

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.

[30]

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.

[31]

R. Wang, Y. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019) 4091–4126. doi: 10.3934/dcds.2019165.

[32]

R. WangL. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R}^N $, Nonlinearity, 32 (2019), 4524-4556.  doi: 10.1088/1361-6544/ab32d7.

[33]

R. Wang and B. Wang, Asymptotic behavior of non-autonomous fractional p-Laplacian equations driven by additive noise on unbounded domains, Bull. Math. Sci., 11 (2021), 50pp. doi: 10.1142/S1664360720500204.

[34]

R. Wang and B. Wang, Asymptotic behavior of non-autonomous fractional stochastic p-Laplacian equations, Comput. Math. Appl., 78 (2019), 3527-3543.  doi: 10.1016/j.camwa.2019.05.024.

[35]

R. Wang and B. Wang, Random dynamics of non-autonomous fractional stochastic p-Laplacian equations on $\mathbb{R}^N $, Banach J. Math. Anal., 15 (2021), 42pp. doi: 10.1007/s43037-020-00107-5.

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