Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation

Linfang Liu; Xianlong Fu

doi:10.3934/cpaa.2017023

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2017, Volume 6: 443-474. Doi: 10.3934/cpaa.2017023

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Existence and upper semicontinuity of (L², L^q) pullback attractors for a stochastic p-laplacian equation

Linfang Liu and
Xianlong Fu^,

Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, China

Author Bio: liulinfang1988@163.com (L. Liu)
xlfu@math.ecnu.edu.cn (X. Fu, the corresponding author)
xlfu@math.ecnu.edu.cn (X. Fu, the corresponding author)

Received: February 2016

Revised: October 2016

Published: March 2017

This research is supported by NSFof China (Nos. 11671142 and11371087), Science and Technology Commission of Shanghai Municipality (No. 13dz2260400) and Shanghai Leading Academic Discipline Project (No. B407), respectively

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Abstract

In this paper we study the dynamical behavior of solutions for a non-autonomous $p$-Laplacian equation driven by a white noise term. We first establish the abstract results on existence and continuity of bi-spatial pullback random attractors for a cocycle. Then by conducting some tail estimates and applying the obtained abstract results we show the existence and upper semi-continuity of $(L^{2}(\mathbb{R}^{n}), L^{q}(\mathbb{R}^{n}))$-pullback attractors for this $p$-Laplacian equation.

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Mathematics Subject Classification: 35B40, 35R60, 37L55, 60H15.

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References

[1]	C. Anh, T. Bao and N. Thanh, Regularity of random attractors for stochastic semi-linear degenerate parabolic equations, Electr. J. Diff. Equ., 207 (2012), 1-25.
[2]	L. Arnold, Random Dynamical Systems, Spring-Verlag, New-York, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[3]	J. Ball, Continuity properties and global attractors of gernerlized semiflows and the NaiverStokes equations, J. Nonl. Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037.
[4]	E. Benedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[5]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Prob. Th. Re. Fields., 100 (1994), 365-393. doi: 10.1007/BF01193705.
[6]	G. Chen, Uniform attractors for the non-autonomous parabolic equation with nonlinear Laplacian principal part in unbounded domain, Diff. Equ. Appl., 2 (2010), 105-121. doi: 10.7153/dea-02-08.
[7]	A. Carvalho and J. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds, J. Diff. Equ., 233 (2007), 622-653. doi: 10.1016/j.jde.2006.08.009.
[8]	A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamcia Systems, Appl. Math. Sciences, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.
[9]	T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonl. Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[10]	T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-NavierStokes equations in some unbounded domains, CR. Acad. Sci. Pari. Ser., 342 (2006), 263-268. doi: 10.1016/j.crma.2005.12.015.
[11]	V. Chepyzhov and M. Vishik, Attractors of non-autonomous dynamical systems and their dimensions, J. Math. Pures. Appl., 73 (1994), 279-333.
[12]	K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.
[13]	B. Gess, Random attractors for singular stochastic evolution equations, J. Diff. Equ., 255 (2013), 524-559. doi: 10.1016/j.jde.2013.04.023.
[14]	K. Wiesenfeld and F. Moss, Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs, Nature, 373 (1995), 33-35.
[15]	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.
[16]	A. Krause, M. 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.
[17]	A. Krause, 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.
[18]	Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Diff. Equ., 245 (2008), 1775-1800. doi: 10.1016/j.jde.2008.06.031.
[19]	Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Diff. Equ., 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.
[20]	J. Li, Y. Li and H. Cui, Existence and upper semicontinuity of random attractors for stochastic p-Laplaican equations on unbounded domains, Electr. J. Diff. Equ., 2014 (2014), 1-27.
[21]	G. Lukaszewicz and A. Tarasinska, On H¹-pullback attractors for non-autonomous micropolar fluid equations in a bounded domains, Nonl. Anal., 71 (2009), 782-788. doi: 10.1016/j.na.2008.10.124.
[22]	H. Li, Y. You and J. Tu, Random attractors and averging for non-autonomous stochastic wave equations with nonlinear damping, J. Diff. Equ., 258 (2015), 148-190. doi: 10.1016/j.jde.2014.09.007.
[23]	Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comp., 190 (2007), 1020-1029. doi: 10.1016/j.amc.2006.11.187.
[24]	J. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Univ. Press, 2001. doi: 10.1007/978-94-010-0732-0.
[25]	B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour, (1992), 185-192.
[26]	J. Simsen and E. Junior, Existence and upper semicontinuity of global attractors for a pLaplacian inclusion, Bol. Soc. Paran. Mat., 1 (2015), 235-245.
[27]	J. Simsen, M. Nascimento and M. 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.
[28]	C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonl. Anal., 63 (2005), 49-65. doi: 10.1016/j.na.2005.04.034.
[29]	R. Temman, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringVerlag, New York, 1998. doi: 10.1007/978-1-4684-0313-8.
[30]	H. Tuckwell, Introduction to Theoretical Neurobiology: Nonlinear and Stochastic Theories, Cambridge University Press, Cambridge, 1998.
[31]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical system, J. Diff. Equ., 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.
[32]	B. Wang, Existence and Upper Semicontinuity of Attractors for Stochastic Equations with Deterministic Non-autonomous Terms, Stoch. Dynam., 14 (2014), 1-31. doi: 10.1142/S0219493714500099.
[33]	B. Wang and L. Guo, Asymptotic behavior of non-autonomous stochastic parabolic equations with nonliear laplacian principal part, Electr. J. of Diff. Equ., 2013 (2013), 1-25.
[34]	Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Diff. Equ., 259 (2015), 728-776. doi: 10.1016/j.jde.2015.02.026.
[35]	M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a p-Laplacian equation in $\mathbb{R}.{n}$, Nonl. Anal., 66 (2007), 1-13. doi: 10.1016/j.na.2005.11.004.
[36]	Y. You, Random dynamics of stochastic reaction-diffusion systems with additive noise, J. Dyn. Diff. Equ., DOI: 10.1007/s10884-015-9431-4, in press, (2015). doi: 10.1007/s10884-015-9431-4.
[37]	W. Zhao and R. Li, (L², L^p)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonl. Anal., 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050.
[38]	C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Diff. Equ., 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.