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Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations

This work is partially supported by NSF of Chongqing Grant No. cstc2018jcyjA0897, the FRF for the Central Universities Grant No. XDJK2020B049 and K.C. Wong Education Foundation and DAAD

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  • In this paper, we obtain the existence and uniqueness of weak pullback mean random attractors for non-autonomous deterministic $ p $-Laplacian equations with random initial data and non-autonomous stochastic $ p $-Laplacian equations with general diffusion terms in Bochner spaces, respectively.

    Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.

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  • [1] L. Arnold, Random Dynamical Systems, 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] W.-J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Commun. Partial Differ. Equ., 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.
    [4] T. CaraballoM. J. Garrido-Atienza and B. Schmalfuss, Existence of exponetially attracting stationary solutions for delay evolution equations, Discrete Contin. Dyn. Syst., 18 (2007), 271-293.  doi: 10.3934/dcds.2007.18.271.
    [5] T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.
    [6] T. CaraballoJ. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.
    [7] I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.  doi: 10.1080/1468936042000207792.
    [8] H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Equat., 9 (1997), 307-341.  doi: 10.1007/BF02219225.
    [9] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.
    [10] J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151. 
    [11] B. Fehrman and B. Gess, Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise, Arch. Rational Mech. Anal., 233 (2019), 249–322. doi: 10.1007/s00205-019-01357-w.
    [12] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.
    [13] H. GaoM. J. Garrido-Atienza and B. Schmalfuss, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309.  doi: 10.1137/130930662.
    [14] M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473–493. doi: 10.3934/dcdsb.2010.14.473.
    [15] M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dyn. Diff. Equat., 23 (2011), 671-681.  doi: 10.1007/s10884-011-9222-5.
    [16] M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parametes $H\in (1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625–654. doi: 10.1137/15M1030303.
    [17] 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.
    [18] B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Diff. Equat., 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.
    [19] B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.
    [20] A. Gu, Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5737–5767. doi: 10.3934/dcdsb.2019104.
    [21] A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.
    [22] 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.
    [23] 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. doi: 10.3934/dcdsb.2018072.
    [24] J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882.  doi: 10.3934/dcds.2009.24.855.
    [25] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A. Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.
    [26] P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.
    [27] 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.
    [28] P. Lindqvist, Notes on the $p$-Laplace Equation, 2006. Available from: http://www.math.ntnu.no/ lqvist/p-laplace.pdf.
    [29] J.-L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.
    [30] K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Diff. Equat., 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.
    [31] Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.
    [32] J. Málek, J. Nečas, M. Rokyta and M. R$\rm\mathring{u}$zička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman & Hall, London, 1996.
    [33] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 2007.
    [34] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185-192, Dresden, 1992.
    [35] Z. ShenS. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.
    [36] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, 1997.
    [37] 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.
    [38] B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on ${\mathbb R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.
    [39] 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.
    [40] B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 1–31. doi: 10.1142/S0219493714500099.
    [41] B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dyn. Diff. Equat., 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.
    [42] B. Wang, Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.  doi: 10.1090/proc/14356.
    [43] B. Wang and B. Guo, Asymptotic behavior of non-autonomous stochastic parabolic equations with nonlinear Laplacian principal part, Electron. J. Differ. Eq., (2013), No. 191, 25 pp.
    [44] S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in ${\mathbb R}^3$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.
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