Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 37L30; Secondary: 35B40, 35B41.

 Citation:

•  [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.