# American Institute of Mathematical Sciences

December  2016, 9(6): 1939-1957. doi: 10.3934/dcdss.2016079

## Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations

 1 School of Mathematics and Statistics, Southwest University, Chongqing 400715

Received  June 2015 Revised  August 2016 Published  November 2016

A global attractor in $L^2$ is shown for weakly dissipative $p$-Laplace equations on the entire Euclid space, where the weak dissipativeness means that the order of the source is lesser than $p-1$. Half-time decomposition and induction techniques are utilized to present the tail estimate outside a ball. It is also proved that the equations in both strongly and weakly dissipative cases possess an $(L^2,L^r)$-attractor for $r$ belonging to a special interval, which contains the critical exponent $p$. The obtained attractor is proved to be approximated by the corresponding attractor inside a ball in the sense of upper strictly and lower semicontinuity.
Citation: Yangrong Li, Jinyan Yin. Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1939-1957. doi: 10.3934/dcdss.2016079
##### References:
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Li, The asymptotic behavior of the stochastic Ginzburg-Landou equation with additive noise,, Appl. Math. Comput., 198 (2008), 849. doi: 10.1016/j.amc.2007.09.029. Google Scholar [26] Z. Wang and S. Zhou, Random attractors for stochastic reaction-diffusion equations with multiplicative noise on unbounded domains,, J. Math. Anal. Appl., 384 (2011), 160. doi: 10.1016/j.jmaa.2011.02.082. Google Scholar [27] M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation,, J. Math. Anal. Appl., 327 (2007), 1130. doi: 10.1016/j.jmaa.2006.04.085. Google Scholar [28] X. Yan and C. Zhong, $L^p$-uniform attractor for nonautonomous reaction-diffusion equations in unbounded domains,, J. Math. Phys., 49 (2008). doi: 10.1063/1.3000575. Google Scholar [29] J. Y. Yin, Y. R. Li and H. J. Zhao, Random attractors for stochastic semi-linear degenerate parabolic equations with addtive noise in $L^q$,, Appl. Math. Comput., 225 (2013), 526. doi: 10.1016/j.amc.2013.09.051. Google Scholar [30] W. 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Equ., 223 (2006), 367. doi: 10.1016/j.jde.2005.06.008. Google Scholar

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##### References:
 [1] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, J. Differ. Equ., 246 (2009), 845. doi: 10.1016/j.jde.2008.05.017. Google Scholar [2] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems,, International J. Bifur. Chaos, 11 (2001), 143. doi: 10.1142/S0218127401002031. Google Scholar [3] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractor for Infinite-dimensional Non-autonomous Dynamical Systems,, Appl. Math. Sciences, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar [4] B. Gess, W. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise,, J. Differ. Equ., 251 (2011), 1225. doi: 10.1016/j.jde.2011.02.013. Google Scholar [5] B. Gess, Random attractors for degenerate stochastic partial differential equations,, J. Dyn. Differ. Equ., 25 (2013), 121. doi: 10.1007/s10884-013-9294-5. Google Scholar [6] B. 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Wang, Dynamics of the non-autonomous stochastic p-Laplace equation driven by multiplicative noise,, Appl. Math. Comput., 246 (2014), 365. doi: 10.1016/j.amc.2014.08.033. Google Scholar [12] A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains,, J. Math. Anal. Appl., 417 (2014), 1018. doi: 10.1016/j.jmaa.2014.03.037. Google Scholar [13] J. Li, Y. R. Li and B. Wang, Random attractors of reaction-diffusion equations with multiplicative noise in $L^p$,, Appl. Math. Comput., 215 (2010), 3399. doi: 10.1016/j.amc.2009.10.033. Google Scholar [14] J. Li, Y. R. Li and H. Y. Cui, Existence and upper semicontinuity of random attractors for stochastic p-Laplacian equations on unbounded domains,, Electronic J. Differ. Equ., 2014 (2014), 1. Google Scholar [15] Y. R. Li, A. H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations,, J. Differ. Equ., 258 (2015), 504. doi: 10.1016/j.jde.2014.09.021. Google Scholar [16] Y. R. Li, H. Y. Cui and J. Li, Upper semi-continuouity and regularity of random attractors on p-times integrable spaces and applications,, Nonlinear Anal. TMA, 109 (2014), 33. doi: 10.1016/j.na.2014.06.013. Google Scholar [17] Y. R. Li and B. L. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations,, J. Differ. Equ., 245 (2008), 1775. doi: 10.1016/j.jde.2008.06.031. Google Scholar [18] Y. R. Li and J. Y. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations,, Discrete Contin. Dyn. Syst. B, 21 (2016), 1203. doi: 10.3934/dcdsb.2016.21.1203. Google Scholar [19] T. F. Ma and M. L. Pelicer, Attractors for weakly damped beam equations with p-Laplacian,, Discrete Contin. Dyn. Sys., (2013), 525. doi: 10.3934/proc.2013.2013.525. Google Scholar [20] J. Simsen, A note on p-Laplacian parabolic problems in R-n,, Nonliear Anal. TMA, 75 (2012), 6620. doi: 10.1016/j.na.2012.08.007. Google Scholar [21] J. Simsen, M. 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. doi: 10.1016/j.jmaa.2013.12.019. Google Scholar [22] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,, J. Differ. Equ., 253 (2012), 1544. doi: 10.1016/j.jde.2012.05.015. Google Scholar [23] B. Wang and B. Guo, Asymptotic behavior of non-autonomous stochastic equations with nonlinear Laplacian principal part,, Electronic J. Differ. Equ., 191 (2013), 1. Google Scholar [24] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Second ed., (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar [25] G. L. Wang, B. L.Guo and Y. R. Li, The asymptotic behavior of the stochastic Ginzburg-Landou equation with additive noise,, Appl. Math. Comput., 198 (2008), 849. doi: 10.1016/j.amc.2007.09.029. Google Scholar [26] Z. Wang and S. Zhou, Random attractors for stochastic reaction-diffusion equations with multiplicative noise on unbounded domains,, J. Math. Anal. Appl., 384 (2011), 160. doi: 10.1016/j.jmaa.2011.02.082. Google Scholar [27] M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation,, J. Math. Anal. Appl., 327 (2007), 1130. doi: 10.1016/j.jmaa.2006.04.085. Google Scholar [28] X. Yan and C. Zhong, $L^p$-uniform attractor for nonautonomous reaction-diffusion equations in unbounded domains,, J. Math. Phys., 49 (2008). doi: 10.1063/1.3000575. Google Scholar [29] J. Y. Yin, Y. R. Li and H. J. Zhao, Random attractors for stochastic semi-linear degenerate parabolic equations with addtive noise in $L^q$,, Appl. Math. Comput., 225 (2013), 526. doi: 10.1016/j.amc.2013.09.051. Google Scholar [30] W. Q. Zhao, Regularity of random attractors for a degenerate parabolic equations driven by additive noise,, Appl. Math. Comput., 239 (2014), 358. doi: 10.1016/j.amc.2014.04.106. Google Scholar [31] W. Zhao and Y. R. Li, $(L^2,L^p)$-random attractors for stochastic reaction-diffusion on unbounded domains,, Nonlinear Anal. TMA, 75 (2012), 485. doi: 10.1016/j.na.2011.08.050. Google Scholar [32] W. Q. Zhao and Y. R. Li, Existence of random attractors for a $p$-Laplacian-type equation with additive noise,, Abstr. Appl. Anal., 10 (2011). doi: 10.1155/2011/616451. Google Scholar [33] W. Q. Zhao and Y. R. Li, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noises,, Dyn. Partial Differ. Equ., 11 (2014), 269. doi: 10.4310/DPDE.2014.v11.n3.a4. Google Scholar [34] 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. Differ. Equ., 223 (2006), 367. doi: 10.1016/j.jde.2005.06.008. Google Scholar
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