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

[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.

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.

[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, 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.

[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.

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