# American Institute of Mathematical Sciences

March  2013, 12(2): 735-754. doi: 10.3934/cpaa.2013.12.735

## Long-time dynamics of the parabolic $p$-Laplacian equation

 1 Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara, Turkey, Turkey

Received  July 2011 Revised  March 2012 Published  September 2012

In this paper, we study the long-time behaviour of solutions of Cauchy problem for the parabolic $p$-Laplacian equation with variable coefficients. Under the mild conditions on the coefficient of the principal part and without upper growth restriction on the source function, we prove that this problem possesses a compact and invariant global attractor in $L^2(R^n)$.
Citation: Pelin G. Geredeli, Azer Khanmamedov. Long-time dynamics of the parabolic $p$-Laplacian equation. Communications on Pure & Applied Analysis, 2013, 12 (2) : 735-754. doi: 10.3934/cpaa.2013.12.735
##### References:
 [1] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (1988). Google Scholar [2] A. V. Babin and M. I. Vishik, Attractors of differential evolution equations in unbounded domain,, Proc. Roy. Soc. Edinburg, 116A (1990), 221. doi: 10.1017/S0308210500031498. Google Scholar [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992). Google Scholar [4] E. Feireisl, Ph. Laurencot, F. Simondon and H. Toure, Compact attractors for reaction diffusion equations in $R^n$,, C. R. Acad. Sci. Paris Ser. I, 319 (1994), 147. Google Scholar [5] B. Wang, Attractors for reaction diffusion equations in unbounded domains,, Physica D, 128 (1999), 41. Google Scholar [6] M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain,, Comm. Pure Appl. Math., 54 (2001), 625. doi: 10.1002/cpa.1011. Google Scholar [7] J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains,, Nonlinear Analysis: Theory, 56 (2004). doi: 10.1016/j.na.2003.09.023. Google Scholar [8] A. N. Carvalho, J. W. Cholewa and T. Dlotko, Global attractors for problems with monotone operators,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 2 (1999), 693. Google Scholar [9] A. N. Carvalho and C. B. Gentile, Asymptotic behavior of non-linear parabolic equations with monotone principial part,, J. Math. Anal. Appl., 280 (2003), 252. doi: 10.1016/S0022-247X(03)00037-4. Google Scholar [10] M. Nakao and N. Aris, On global attractor for nonlinear parabolic equation of $m$-Laplacian type,, J. Math. Anal. Appl., 331 (2007), 793. doi: 10.1016/j.jmaa.2006.08.044. Google Scholar [11] M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation,, J. Math. Anal. Appl., 337 (2007), 1130. doi: 10.1016/j.jmaa.2006.04.085. Google Scholar [12] M. Nakao and C. Chen, On global attractor for a nonlinear parabolic equation of $m$-Laplacian type in $R^n$,, Funkcialaj Ekvacioj, 50 (2007), 449. doi: 10.1619/fesi.50.449. Google Scholar [13] C. Chen, L. Shi and H. Wang, Existence of a global attractors in $L^p$ for $m$-Laplacian parabolic equation in $R^n$,, Boundary Value Problems, 2009 (2009), 1. doi: 10.1155/2009/563767. Google Scholar [14] A. Kh. 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. doi: 10.1016/j.jmaa.2005.05.003. Google Scholar [15] A. Kh. Khanmamedov, Global attractors for one dimensional $p$-Laplacian equation,, Nonlinear Analysis: Theory, 71 (2009), 155. doi: 10.1016/j.na.2008.10.037. Google Scholar [16] M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a $p$-Laplacian equation in $R^n$,, Nonlinear Analysis: Theory, 66 (2007), 1. doi: 10.1016/j.na.2005.11.004. Google Scholar [17] C. T. Anh and T. D. Ke, Long time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators,, Nonlinear Analysis: Theory, 71 (2009), 4415. doi: 10.1016/j.na.2009.02.125. Google Scholar [18] C. T. Anh and T. D. Ke, On quasilinear parabolic equations involving weighted p-Laplacian operators,, Nonlinear Differential Equations and Applications, 17 (2010), 195. doi: 10.1007/s00030-009-0048-3. Google Scholar [19] A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement-dependent damping,, Math. Methods Appl. Sci., 33 (2010), 177. doi: 10.1002/mma.1161. Google Scholar [20] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Mathematical Surveys Monographs, (1997). Google Scholar [21] J. Simon, Compact sets in the space $L_p(0, T;B)$,, Annali Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [22] M. A. Krasnoselskii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,", P. Noordhoff Ltd., (1961). Google Scholar [23] J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications,", \textbf{1}, 1 (1972). Google Scholar [24] O. A. Ladyzhenskaya, On the determination of minimal global attractors for the Navier-Stokes equations and other partial differential equations,, Uspekhi Mat. Nauk, 42 (1987), 27. doi: 10.1070/RM1987v042n06ABEH001503. Google Scholar

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##### References:
 [1] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (1988). Google Scholar [2] A. V. Babin and M. I. Vishik, Attractors of differential evolution equations in unbounded domain,, Proc. Roy. Soc. Edinburg, 116A (1990), 221. doi: 10.1017/S0308210500031498. Google Scholar [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992). Google Scholar [4] E. Feireisl, Ph. Laurencot, F. Simondon and H. Toure, Compact attractors for reaction diffusion equations in $R^n$,, C. R. Acad. Sci. Paris Ser. I, 319 (1994), 147. Google Scholar [5] B. Wang, Attractors for reaction diffusion equations in unbounded domains,, Physica D, 128 (1999), 41. Google Scholar [6] M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain,, Comm. Pure Appl. Math., 54 (2001), 625. doi: 10.1002/cpa.1011. Google Scholar [7] J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains,, Nonlinear Analysis: Theory, 56 (2004). doi: 10.1016/j.na.2003.09.023. Google Scholar [8] A. N. Carvalho, J. W. Cholewa and T. Dlotko, Global attractors for problems with monotone operators,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 2 (1999), 693. Google Scholar [9] A. N. Carvalho and C. B. Gentile, Asymptotic behavior of non-linear parabolic equations with monotone principial part,, J. Math. Anal. Appl., 280 (2003), 252. doi: 10.1016/S0022-247X(03)00037-4. Google Scholar [10] M. Nakao and N. Aris, On global attractor for nonlinear parabolic equation of $m$-Laplacian type,, J. Math. Anal. Appl., 331 (2007), 793. doi: 10.1016/j.jmaa.2006.08.044. Google Scholar [11] M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation,, J. Math. Anal. Appl., 337 (2007), 1130. doi: 10.1016/j.jmaa.2006.04.085. Google Scholar [12] M. Nakao and C. Chen, On global attractor for a nonlinear parabolic equation of $m$-Laplacian type in $R^n$,, Funkcialaj Ekvacioj, 50 (2007), 449. doi: 10.1619/fesi.50.449. Google Scholar [13] C. Chen, L. Shi and H. Wang, Existence of a global attractors in $L^p$ for $m$-Laplacian parabolic equation in $R^n$,, Boundary Value Problems, 2009 (2009), 1. doi: 10.1155/2009/563767. Google Scholar [14] A. Kh. 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. doi: 10.1016/j.jmaa.2005.05.003. Google Scholar [15] A. Kh. Khanmamedov, Global attractors for one dimensional $p$-Laplacian equation,, Nonlinear Analysis: Theory, 71 (2009), 155. doi: 10.1016/j.na.2008.10.037. Google Scholar [16] M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a $p$-Laplacian equation in $R^n$,, Nonlinear Analysis: Theory, 66 (2007), 1. doi: 10.1016/j.na.2005.11.004. Google Scholar [17] C. T. Anh and T. D. Ke, Long time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators,, Nonlinear Analysis: Theory, 71 (2009), 4415. doi: 10.1016/j.na.2009.02.125. Google Scholar [18] C. T. Anh and T. D. Ke, On quasilinear parabolic equations involving weighted p-Laplacian operators,, Nonlinear Differential Equations and Applications, 17 (2010), 195. doi: 10.1007/s00030-009-0048-3. Google Scholar [19] A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement-dependent damping,, Math. Methods Appl. Sci., 33 (2010), 177. doi: 10.1002/mma.1161. Google Scholar [20] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Mathematical Surveys Monographs, (1997). Google Scholar [21] J. Simon, Compact sets in the space $L_p(0, T;B)$,, Annali Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [22] M. A. Krasnoselskii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,", P. Noordhoff Ltd., (1961). Google Scholar [23] J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications,", \textbf{1}, 1 (1972). Google Scholar [24] O. A. Ladyzhenskaya, On the determination of minimal global attractors for the Navier-Stokes equations and other partial differential equations,, Uspekhi Mat. Nauk, 42 (1987), 27. doi: 10.1070/RM1987v042n06ABEH001503. Google Scholar
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