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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 |
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.
References:
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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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