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March  2017, 6(2): 443-474. doi: 10.3934/cpaa.2017023

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

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

xlfu@math.ecnu.edu.cn (X. Fu, the corresponding author)

Received  February 2016 Revised  October 2016 Published  January 2017

Fund Project: 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.

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.

Citation: Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation. Communications on Pure & Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023
##### References:
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Appl., 2 (2010), 105-121. doi: 10.7153/dea-02-08.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] V. Chepyzhov and M. Vishik, Attractors of non-autonomous dynamical systems and their dimensions, J. Math. Pures. Appl., 73 (1994), 279-333. Google Scholar [12] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.  doi: 10.1007/978-3-662-00547-7.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] R. Temman, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringVerlag, New York, 1998.  doi: 10.1007/978-1-4684-0313-8.  Google Scholar [30] H. Tuckwell, Introduction to Theoretical Neurobiology: Nonlinear and Stochastic Theories, Cambridge University Press, Cambridge, 1998.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [37] W. Zhao and R. Li, (L2, Lp)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonl. Anal., 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [2] L. Arnold, Random Dynamical Systems, Spring-Verlag, New-York, Berlin, 1998.  doi: 10.1007/978-3-662-12878-7.  Google Scholar [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.  Google Scholar [4] E. Benedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.  doi: 10.1007/978-1-4612-0895-2.  Google Scholar [5] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Prob. Th. Re. Fields., 100 (1994), 365-393. doi: 10.1007/BF01193705.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] V. Chepyzhov and M. Vishik, Attractors of non-autonomous dynamical systems and their dimensions, J. Math. Pures. Appl., 73 (1994), 279-333. Google Scholar [12] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.  doi: 10.1007/978-3-662-00547-7.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] R. Temman, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringVerlag, New York, 1998.  doi: 10.1007/978-1-4684-0313-8.  Google Scholar [30] H. Tuckwell, Introduction to Theoretical Neurobiology: Nonlinear and Stochastic Theories, Cambridge University Press, Cambridge, 1998.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [37] W. Zhao and R. Li, (L2, Lp)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonl. Anal., 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050.  Google Scholar [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.  Google Scholar
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