American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020290

Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay

Pengyu Chen , and  Xuping Zhang

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

* Corresponding author: Pengyu Chen

Received  July 2020 Published  October 2020

Fund Project: Chen was supported by the National Natural Science Foundation of China (No. 12061063), Project of NWNU-LKQN2019-3 and China Scholarship Council (No. 201908625016). Zhang was supported by 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 asymptotic behavior of the solutions to a class of non-autonomous nonlocal fractional stochastic parabolic equations with delay defined on bounded domain. We first prove the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise intensity. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform estimates of solutions in fractional Sobolev space $ H^\alpha(\mathbb{R}^n) $ with $ \alpha\in (0,1) $ as well as their time derivatives in $ L^2(\mathbb{R}^n) $. Finally, we establish the upper semi-continuity of the random attractors when noise intensity and time delay approaches zero, respectively.

Keywords: Random attractors, fractional Laplacian, stochastic delay equations, upper semi-continuity, asymptotical compactness.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.
Citation: Pengyu Chen, Xuping Zhang. Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020290
References:
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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.  Google Scholar

[2]

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

[3]

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

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

L. A. Caffarelli, J.-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.  Google Scholar

[7]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar

[8]

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

[9]

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

[10]

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[11]

T. Caraballo, M. J. Garrido-Atienza, B. 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.  Google Scholar

[12]

P. Chen, Y. Li and X. Zhang, Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families, Discrete Contin. Dyn. Syst. Ser. B, published online, 2020. doi: 10.3934/dcdsb.2020171.  Google Scholar

[13]

P. Chen, X. Zhang and Y. Li, Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fract. Calcu. Appl. Anal., 23 (2020), 268-291.  doi: 10.1515/fca-2020-0011.  Google Scholar

[14]

Z. Chen and B. Wang, Invariant measures of fractional stochastic delay reaction-diffusion equations on unbounded domains, submitted. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

A. Gu, D. Li, B. 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.  Google Scholar

[22]

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

[23]

X. Han, P. E. Kloden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813.  Google Scholar

[24]

J. Huang, T. 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.  Google Scholar

[25]

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.  Google Scholar

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

D. Li, K. Lu, B. 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.  Google Scholar

[29]

D. Li, L. Shi and X. Wang, Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5121-5148.  doi: 10.3934/dcdsb.2019046.  Google Scholar

[30]

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.  Google Scholar

[31]

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.  Google Scholar

[32]

H. Lu, P. W. Bates, S. Lü and M. Zhang, Dynamics of 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.  Google Scholar

[33]

H. Lu, P. W. Bates, S. Lü 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.   Google Scholar

[34]

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

[35]

H. Lu, J. Qi, B. 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.  Google Scholar

[36]

X. Mao, Stochastic Differential Equations and Applications, Second Edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[37]

S. E. A. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, 99, Pitman, Boston, 1984.  Google Scholar

[38]

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.  Google Scholar

[39]

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.  Google Scholar

[40]

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

[41]

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.  Google Scholar

[42]

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

[43]

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.  Google Scholar

[44]

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

[45]

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.  Google Scholar

[46]

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

[47]

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