March  2022, 27(3): 1695-1724. doi: 10.3934/dcdsb.2021107

Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise

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

Received  January 2021 Published  March 2022 Early access  March 2021

Fund Project: This research was supported by Natural Science Foundation of Gansu Province (No. 20JR5RA522), Science Research Project for Colleges and Universities of Gansu Province (No. 2019B-047), Project of NWNU-LKQN2019-13 and Doctoral Research Fund of Northwest Normal University (No. 6014/0002020209)

This paper is concerned with the pullback random attractors of nonautonomous nonlocal fractional stochastic $ p $-Laplacian equation with delay driven by multiplicative white noise defined on bounded domain. We first prove the existence of a continuous nonautonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise. We then show pullback asymptotical compactness of solutions and the existence of tempered random attractors by utilizing the Arzela-Ascoli theorem and appropriate uniform estimates of the solutions. Finally, we establish the upper semicontinuity of the random attractors when time delay approaches zero.

Citation: Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1695-1724. doi: 10.3934/dcdsb.2021107
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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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L. A. CaffarelliJ.-M. 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.

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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.

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T. Caraballo and A. M. Márquez-Durán, Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay, Dyn. Partial Differ. Equ., 10 (2013), 267-281.  doi: 10.4310/DPDE.2013.v10.n3.a3.

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T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Attractors for a random evolution equation with infinite memory: Theoretical results, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1779-1800.  doi: 10.3934/dcdsb.2017106.

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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.

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J. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.

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P. G. Geredeli and A. Khanmamedov, Long-time dynamics of the parabolic $p$-Laplacian equation, Commun. Pure Appl. Anal., 12 (2013), 735-754.  doi: 10.3934/cpaa.2013.12.735.

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

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[21]

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.

[22]

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.

[23]

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[27]

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[28]

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[29]

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.

[30]

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.

[31]

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.

[32]

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.

[33]

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.

[34]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.

[35]

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.

[36]

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.

[37]

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.

[38]

L. Wang and D. Xu, Asymptotic behavior of a class of reaction-diffusion equations with delays, J. Math. Anal. Appl., 281 (2003), 439-453.  doi: 10.1016/S0022-247X(03)00112-4.

[39]

R. WangY. 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.

[40]

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.

[41]

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[42]

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), No. 19, 42 pp. doi: 10.1007/s43037-020-00107-5.

[43]

X. WangK. Lu and B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.

[44]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[45]

M. Warma, On a fractional $(s, p)$-Dirichlet-to-Neumann operator on bounded lipschitz domains, J. Elliptic Parabol. Equ., 4 (2018), 223-269.  doi: 10.1007/s41808-018-0017-2.

[46]

F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1715-1734.  doi: 10.3934/dcdsb.2013.18.1715.

[47]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[48]

W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 18 pp. doi: 10.1063/1.3319566.

[49]

W. Zhao, Long-time random dynamics of stochastic parabolic $p$-Laplacian equations on $\mathbb{R}^N$, Nonlinear Anal., 152 (2017), 196-219.  doi: 10.1016/j.na.2017.01.004.

[50]

W. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.  doi: 10.1016/j.jmaa.2017.06.025.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[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]

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.

[4]

L. A. CaffarelliJ.-M. 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.

[5]

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.

[6]

T. Caraballo and A. M. Márquez-Durán, Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay, Dyn. Partial Differ. Equ., 10 (2013), 267-281.  doi: 10.4310/DPDE.2013.v10.n3.a3.

[7]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Attractors for a random evolution equation with infinite memory: Theoretical results, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1779-1800.  doi: 10.3934/dcdsb.2017106.

[8]

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.

[9]

J. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.

[10]

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.

[11]

P. G. Geredeli and A. Khanmamedov, Long-time dynamics of the parabolic $p$-Laplacian equation, Commun. Pure Appl. Anal., 12 (2013), 735-754.  doi: 10.3934/cpaa.2013.12.735.

[12]

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

[13]

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.

[14]

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

[15]

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.

[16]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[17]

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.

[18]

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.

[19]

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

[20]

P. E. Kloeden and T. Lorenz, Pullback attractors of reaction-diffusion inclusions with space-dependent delay, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1909-1964.  doi: 10.3934/dcdsb.2017114.

[21]

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.

[22]

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.

[23]

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.

[24]

D. LiK. LuB. Wang and X. Wang, Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 39 (2019), 3717-3747.  doi: 10.3934/dcds.2019151.

[25]

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), 072704, 21 pp. doi: 10.1063/1.5063840.

[26]

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.

[27]

H. LuJ. QiB. Wang and M. Zhang, Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683-706.  doi: 10.3934/dcds.2019028.

[28]

S. E. A. Mohammed, Stochastic Functional Differential Equations, Res. Notes in Math. 99, Pitman, Boston, 1984.

[29]

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.

[30]

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.

[31]

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.

[32]

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.

[33]

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.

[34]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.

[35]

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.

[36]

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.

[37]

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.

[38]

L. Wang and D. Xu, Asymptotic behavior of a class of reaction-diffusion equations with delays, J. Math. Anal. Appl., 281 (2003), 439-453.  doi: 10.1016/S0022-247X(03)00112-4.

[39]

R. WangY. 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.

[40]

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.

[41]

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.

[42]

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), No. 19, 42 pp. doi: 10.1007/s43037-020-00107-5.

[43]

X. WangK. Lu and B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.

[44]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[45]

M. Warma, On a fractional $(s, p)$-Dirichlet-to-Neumann operator on bounded lipschitz domains, J. Elliptic Parabol. Equ., 4 (2018), 223-269.  doi: 10.1007/s41808-018-0017-2.

[46]

F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1715-1734.  doi: 10.3934/dcdsb.2013.18.1715.

[47]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[48]

W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 18 pp. doi: 10.1063/1.3319566.

[49]

W. Zhao, Long-time random dynamics of stochastic parabolic $p$-Laplacian equations on $\mathbb{R}^N$, Nonlinear Anal., 152 (2017), 196-219.  doi: 10.1016/j.na.2017.01.004.

[50]

W. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.  doi: 10.1016/j.jmaa.2017.06.025.

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