September  2022, 27(9): 5129-5159. doi: 10.3934/dcdsb.2021267

Dynamics of fractional nonclassical diffusion equations with delay driven by additive noise on $ \mathbb{R}^n $

1. 

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

2. 

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

* Corresponding author: Pengyu Chen

Received  June 2021 Revised  September 2021 Published  September 2022 Early access  November 2021

Fund Project: Research supported by National Natural Science Foundations of China (No. 12061063), Natural Science Foundation of Gansu Province (No. 21JR7RA159 and No. 20JR5RA522), Project of NWNU-LKQN2019-3 and Project of NWNU-LKQN2019-13

In this paper, we study the asymptotic behavior of solutions of fractional nonclassical diffusion equations with delay driven by additive noise defined on unbounded domains. We first prove the uniform compactness of pullback random attractors of the equation with respect to noise intensity and time delay, and then establish the upper semi-continuity of these attractors as either noise intensity or time delay approaches zero.

Citation: Pengyu Chen, Bixiang Wang, Xuping Zhang. Dynamics of fractional nonclassical diffusion equations with delay driven by additive noise on $ \mathbb{R}^n $. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 5129-5159. doi: 10.3934/dcdsb.2021267
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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.

[2]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mech., 37 (1980), 265-296.  doi: 10.1007/BF01202949.

[3]

E. C. Aifantis, Gradient nanomechanics: Applications to deformation, fracture, and diffusion in nanopolycrystals, Metall. Mater. Trans. A, 42 (2011), 2985-2998.  doi: 10.1007/s11661-011-0725-9.

[4]

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.

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L. Bai and F. Zhang, Uniform attractors for multi-valued process generated by non-autonomous nonclassical diffusion equations with delay in unbounded domain without uniqueness of solutions, Asymptotic Anal., 94 (2015), 187-210.  doi: 10.3233/ASY-151299.

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J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.

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V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

T. CaraballoM. J. Garrido-AtienzaB. 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.

[16]

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.

[17]

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.

[18]

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.

[19]

T. Caraballo, A. M. Márquez-Durán and F. Rivero, Well-posedness and asymptotic behavior of a nonclassical nonautonomous diffusion equation with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1540021, 11 pp. doi: 10.1142/S0218127415400210.

[20]

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.

[21]

P. ChenY. Li and X. Zhang, Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1531-1547.  doi: 10.3934/dcdsb.2020171.

[22]

P. Chen and X. Zhang, Existence of attractors for stochastic diffusion equations with fractional damping and time-varying delay, J. Math. Phys., 62 (2021), 022705, 23 pp. doi: 10.1063/5.0022078.

[23]

P. ChenX. 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.

[24]

P. ChenX. Zhang and Y. Li, Cauchy problem for fractional non-autonomous evolution equations, Banach J. Math. Anal., 14 (2020), 559-584.  doi: 10.1007/s43037-019-00008-2.

[25]

P. ChenX. Zhang and Y. Li, Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ. Oper. Appl., 10 (2019), 955-973.  doi: 10.1007/s11868-018-0257-9.

[26]

P. ChenX. Zhang and Y. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control. Syst., 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x.

[27]

S. Cheng, Random attractor for the nonclassical diffusion equation with fading memory, J. Partial Differ. Equ., 28 (2015), 253-268.  doi: 10.4208/jpde.v28.n3.4.

[28]

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

[29]

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. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Commun. Math. Sci., 1 (2003), 133-151.  doi: 10.4310/CMS.2003.v1.n1.a9.

[31]

M. J. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.

[32]

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.

[33]

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