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

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

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

Yangrong Li ^1, and Jinyan Yin ^1,

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

Received June 2015 Revised August 2016 Published November 2016

Abstract
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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.

Keywords: weak dissipativeness, lower semicontinuity of attractors, regularity of attractors, Global attractors, p-Laplace equations, unbounded domains..

Mathematics Subject Classification: Primary: 37L30; Secondary: 35B40, 35B4.

Citation: Yangrong Li, Jinyan Yin. Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1939-1957. doi: 10.3934/dcdss.2016079

References:

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[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), 3399-3407. doi: 10.1016/j.amc.2009.10.033.
[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), 1-27.
[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), 504-534. doi: 10.1016/j.jde.2014.09.021.
[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), 33-44. doi: 10.1016/j.na.2014.06.013.
[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), 1775-1800. doi: 10.1016/j.jde.2008.06.031.
[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), 1203-1223. doi: 10.3934/dcdsb.2016.21.1203.
[19]	T. F. Ma and M. L. Pelicer, Attractors for weakly damped beam equations with p-Laplacian, Discrete Contin. Dyn. Sys., SI, (2013), 525-534. doi: 10.3934/proc.2013.2013.525.
[20]	J. Simsen, A note on p-Laplacian parabolic problems in R-n, Nonliear Anal. TMA, 75 (2012), 6620-6624. doi: 10.1016/j.na.2012.08.007.
[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), 685-699. doi: 10.1016/j.jmaa.2013.12.019.
[22]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.
[23]	B. Wang and B. Guo, Asymptotic behavior of non-autonomous stochastic equations with nonlinear Laplacian principal part, Electronic J. Differ. Equ., 191 (2013), 1-25.
[24]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second ed., Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[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), 849-857. doi: 10.1016/j.amc.2007.09.029.
[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), 160-172. doi: 10.1016/j.jmaa.2011.02.082.
[27]	M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142. doi: 10.1016/j.jmaa.2006.04.085.
[28]	X. Yan and C. Zhong, $L^p$-uniform attractor for nonautonomous reaction-diffusion equations in unbounded domains, J. Math. Phys., 49 (2008), 102705, 17pp. doi: 10.1063/1.3000575.
[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), 526-540. doi: 10.1016/j.amc.2013.09.051.
[30]	W. Q. Zhao, Regularity of random attractors for a degenerate parabolic equations driven by additive noise, Appl. Math. Comput., 239 (2014), 358-374. doi: 10.1016/j.amc.2014.04.106.
[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), 485-502. doi: 10.1016/j.na.2011.08.050.
[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), Article ID 616451, 21pp. doi: 10.1155/2011/616451.
[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), 269-298. doi: 10.4310/DPDE.2014.v11.n3.a4.
[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), 367-399. doi: 10.1016/j.jde.2005.06.008.

show all references

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), 845-869. doi: 10.1016/j.jde.2008.05.017.
[2]	P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, International J. Bifur. Chaos, 11 (2001), 143-153. doi: 10.1142/S0218127401002031.
[3]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractor for Infinite-dimensional Non-autonomous Dynamical Systems, Appl. Math. Sciences, Vol. 182, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.
[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), 1225-1253. doi: 10.1016/j.jde.2011.02.013.
[5]	B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Differ. Equ., 25 (2013), 121-157. doi: 10.1007/s10884-013-9294-5.
[6]	B. Gess, Random attractors for singular stochastic evolution equations, J. Differ. Equ., 255 (2013), 524-559. doi: 10.1016/j.jde.2013.04.023.
[7]	B. Gess, Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise, Annals Probability, 42 (2014), 818-864. doi: 10.1214/13-AOP869.
[8]	A. K. 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.
[9]	A. K. Khanmamedov, Global attractors for one dimensional p-Laplacian equation, Nonlinear Anal. TMA, 71 (2009), 155-171. doi: 10.1016/j.na.2008.10.037.
[10]	P. G. Geredeli and A. Khanmamedov, Long-time dynamics of the parabolic p-Laplacian equation, Commun Pure Appl Anal, 12 (2013), 735-754. doi: 10.3934/cpaa.2013.12.735.
[11]	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.
[12]	A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038. doi: 10.1016/j.jmaa.2014.03.037.
[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), 3399-3407. doi: 10.1016/j.amc.2009.10.033.
[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), 1-27.
[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), 504-534. doi: 10.1016/j.jde.2014.09.021.
[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), 33-44. doi: 10.1016/j.na.2014.06.013.
[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), 1775-1800. doi: 10.1016/j.jde.2008.06.031.
[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), 1203-1223. doi: 10.3934/dcdsb.2016.21.1203.
[19]	T. F. Ma and M. L. Pelicer, Attractors for weakly damped beam equations with p-Laplacian, Discrete Contin. Dyn. Sys., SI, (2013), 525-534. doi: 10.3934/proc.2013.2013.525.
[20]	J. Simsen, A note on p-Laplacian parabolic problems in R-n, Nonliear Anal. TMA, 75 (2012), 6620-6624. doi: 10.1016/j.na.2012.08.007.
[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), 685-699. doi: 10.1016/j.jmaa.2013.12.019.
[22]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.
[23]	B. Wang and B. Guo, Asymptotic behavior of non-autonomous stochastic equations with nonlinear Laplacian principal part, Electronic J. Differ. Equ., 191 (2013), 1-25.
[24]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second ed., Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[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), 849-857. doi: 10.1016/j.amc.2007.09.029.
[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), 160-172. doi: 10.1016/j.jmaa.2011.02.082.
[27]	M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142. doi: 10.1016/j.jmaa.2006.04.085.
[28]	X. Yan and C. Zhong, $L^p$-uniform attractor for nonautonomous reaction-diffusion equations in unbounded domains, J. Math. Phys., 49 (2008), 102705, 17pp. doi: 10.1063/1.3000575.
[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), 526-540. doi: 10.1016/j.amc.2013.09.051.
[30]	W. Q. Zhao, Regularity of random attractors for a degenerate parabolic equations driven by additive noise, Appl. Math. Comput., 239 (2014), 358-374. doi: 10.1016/j.amc.2014.04.106.
[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), 485-502. doi: 10.1016/j.na.2011.08.050.
[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), Article ID 616451, 21pp. doi: 10.1155/2011/616451.
[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), 269-298. doi: 10.4310/DPDE.2014.v11.n3.a4.
[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), 367-399. doi: 10.1016/j.jde.2005.06.008.

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