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Invariant sample measures and random Liouville type theorem for a nonautonomous stochastic $ p $-Laplacian equation

  • *Corresponding author: Jintao Wang

    *Corresponding author: Jintao Wang 

Jintao Wang is supported by NSF of China (No. 11801190) and Xiaoqian Zhang is supported by Graduate Innovation Fund (No. 316202101029) of Wenzhou University in 2021

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  • We introduce invariant sample measures to nonautonomous random dynamical systems, and consider the dynamical behaviors of a nonautonomous stochastic $ p $-Laplacian equation with multiplicative noise on a bounded domain. We first use the asymptotic a priori estimate method to prove the existence of $ (L^2, L^q) $-pullback random attractors for the generated nonautonomous random dynamical system. Then, we establish the existence of invariant sample measures and random Liouville type theorem in $ L^2 $ for this equation. Moreover, the invariant sample measures are carried by $ W_0^{1, p}\cap L^q $.

    Mathematics Subject Classification: Primary: 35R60, 76F20; Secondary: 35B41.


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  • [1] L. Arnold, Random Dynamical Systems, Springer, New York, 1998. doi: 10.1007/978-3-662-12878-7.
    [2] M. ChekrounE. Simonnet and M. Ghil, Stochastic climate dynamics: Random attractors and time-dependent invariant measures, Phys. D, 240 (2011), 1685-1700.  doi: 10.1016/j.physd.2011.06.005.
    [3] G. Chen, Uniform attractors for the non-autonomous parabolic equation with nonlinear Laplacian principal part in unbounded domain, Differ. Equ. Appl., 2 (2010), 105-121.  doi: 10.7153/dea-02-08.
    [4] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781107295513.
    [5] I. EkrenI. Kukavica and M. Ziane, Existence of invariant measures for the stochastic damped Schrödinger equation, Stoch. Partial Differ. Equ. Anal. Comput., 5 (2017), 343-367.  doi: 10.1007/s40072-016-0090-1.
    [6] C. FoiasO. ManleyR. Rosa and  R. TemamNavier-Stokes Equations and Turbulence, Cambridge University Press, Cambrige, 2001.  doi: 10.1017/CBO9780511546754.
    [7] A. 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.
    [8] A. KrauseM. Lewis and B. Wang, Dynamics of the non-autonomous stochastic $p$-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376.  doi: 10.1016/j.amc.2014.08.033.
    [9] J. LiY. Li and H. Cui, Existence and upper semicontinuity of random attractors for stochastic $p$-Laplacian equations on unbounded domains, Electron. J. Differential Equations, 2014 (2014), 1-27. 
    [10] Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equaitons, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.
    [11] J. Lions, Quelques Méthodes de Résolution des Problémes Aux Limites Non Linéaires, Dunod, Paris, 1969.
    [12] L. F. Liu and X. L. Fu, Existence and upper semicontinuity of $(L^2, L^q)$ pullback attractors for a stochastic $p$-laplacian equation, Commun. Pure Appl. Anal., 16 (2017), 443-473.  doi: 10.3934/cpaa.2017023.
    [13] W. Liu, On the stochastic $p$-Laplace equation, J. Math. Anal. Appl., 360 (2009), 737-751.  doi: 10.1016/j.jmaa.2009.07.020.
    [14] Y. Liu, L. Yang and C. Zhong, Asymptotic regularity for $p$-Laplacian equation, J. Math. Phys., 51 (2010), 052702, 7 pp. doi: 10.1063/1.3427318.
    [15] G. ŁukaszewiczJ. Real and R. C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250.  doi: 10.1007/s10884-011-9213-6.
    [16] G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211.
    [17] G. R. Sell and Y. C. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.
    [18] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, 1997. doi: 10.1090/surv/049.
    [19] J. SimsenM. J. D. Nascimento and M. S. Simsen, Existence and upper semicontinuity of pullback attractors for non-autonomous $p$-Laplacian parabolic problems, J. Math. Anal. Appl., 413 (2014), 685-699.  doi: 10.1016/j.jmaa.2013.12.019.
    [20] J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Math. Monogr., Oxford University Press, 2007.
    [21] B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equaitons, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.
    [22] B. X. Wang and B. L. Guo, Asymptotic behavior of non-autonomous stochastic parabolic equations with nonlinear Laplacian principal part, Electron. J. Differential Equations, 2013 (2013), 1-25. 
    [23] J. Wang, C. Li, L. Yang and M. Jia, Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise, J. Math. Phys., 62 (2021), Paper No. 111507, 31 pp. doi: 10.1063/5.0039187.
    [24] J. Wang, X. Zhang and C. Li, Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift-Hohenberg equation, prepint, 2022, arXiv: 2202.13329.
    [25] J. WangX. Zhang and C. Zhao, Statistical solutions for a nonautonomous modified Swift-Hohenberg equation, Math. Methods Appl. Sci., 44 (2021), 14502-14516.  doi: 10.1002/mma.7719.
    [26] J. Wang, C. Zhao and T. Caraballo, Invariant measures for the 3D globally modified Navier-Stokes equations with unbounded variable delays, Commun. Nonlinear Sci. Numer. Simul., 91 (2020), 105459, 14 pp. doi: 10.1016/j.cnsns.2020.105459.
    [27] X. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540.  doi: 10.3934/dcds.2009.23.521.
    [28] X. WangJ. Wang and C. Li, Invariant measures and statistical solutions for a nonautonomous nonlocal Swift-Hohenberg equation, Dyn. Syst., 37 (2022), 136-158.  doi: 10.1080/14689367.2021.2020215.
    [29] K. Wiesenfeld and F. Moss, Stochastic resonance and the benefits of noise: From ice ages to crayfish and squids, Nature, 373 (1995), 33-35. 
    [30] L. YangM. Yang and C. Zhong, Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651. 
    [31] M. YangC. Sun and C. Zhong, Global attractors for $p$-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142.  doi: 10.1016/j.jmaa.2006.04.085.
    [32] J. Yin and Y. Li, Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic $p$-Laplacian equations on ${\mathbb{R}}^n$, Math. Methods Appl. Sci., 40 (2017), 4863-4879. 
    [33] C. ZhaoT. Caraballo and G. Łukaszewicz, Statistical solution and Liouville type theorem for the Klein-Gordon-Schrödinger equations, J. Differential Equations, 281 (2020), 1-32.  doi: 10.1016/j.jde.2021.01.039.
    [34] C. ZhaoJ. Wang and T. Caraballo, Invariant sample measures and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations, J. Differential Equaitons, 317 (2022), 474-494.  doi: 10.1016/j.jde.2022.02.007.
    [35] W. Zhao and Y. Li, $(L^2, L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal., 75 (2012), 485-502.  doi: 10.1016/j.na.2011.08.050.
    [36] K. Zhu, Y. Xie, F. Zhou and Q. Zhou, Pullback attractors for $p$-Laplacian equations with delays, J. Math. Phys., 62 (2021), 022702, 17 pp. doi: 10.1063/1.5126618.
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