# 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 and 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, Springer-Verlag, New York, 1988. [2] A. V. Babin and M. I. Vishik, Attractors of differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburg, 116A (1990), 221-243. doi: 10.1017/S0308210500031498. [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. [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), 147-151. [5] B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. [6] M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011. [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, Methods & Applications, 56 (2004), 515 - 554. doi: 10.1016/j.na.2003.09.023. [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), 693-706. [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), 252-272. doi: 10.1016/S0022-247X(03)00037-4. [10] M. Nakao and N. Aris, On global attractor for nonlinear parabolic equation of $m$-Laplacian type, J. Math. Anal. Appl., 331 (2007), 793-809. doi: 10.1016/j.jmaa.2006.08.044. [11] M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation, J. Math. Anal. Appl., 337 (2007), 1130-1142. doi: 10.1016/j.jmaa.2006.04.085. [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), 449-468. doi: 10.1619/fesi.50.449. [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), 1-17. doi: 10.1155/2009/563767. [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), 601-615. doi: 10.1016/j.jmaa.2005.05.003. [15] A. Kh. Khanmamedov, Global attractors for one dimensional $p$-Laplacian equation, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 155-171. doi: 10.1016/j.na.2008.10.037. [16] M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a $p$-Laplacian equation in $R^n$, Nonlinear Analysis: Theory, Methods and Applications, 66 (2007), 1-13. doi: 10.1016/j.na.2005.11.004. [17] C. T. Anh and T. D. Ke, Long time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 4415-4422. doi: 10.1016/j.na.2009.02.125. [18] C. T. Anh and T. D. Ke, On quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Differential Equations and Applications, 17 (2010), 195-212. doi: 10.1007/s00030-009-0048-3. [19] A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement-dependent damping, Math. Methods Appl. Sci., 33 (2010), 177-187. doi: 10.1002/mma.1161. [20] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Mathematical Surveys Monographs, 49, American Mathematical Society, 1997. [21] J. Simon, Compact sets in the space $L_p(0, T;B)$, Annali Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [22] M. A. Krasnoselskii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces," P. Noordhoff Ltd., Groningen, 1961. [23] J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," 1, Springer-Verlag, New York-Heidelberg, 1972. [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), 25- 60; Russian Math. Surveys, 42 (1987), 27-73 (English Transl.). doi: 10.1070/RM1987v042n06ABEH001503.

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##### References:
 [1] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. [2] A. V. Babin and M. I. Vishik, Attractors of differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburg, 116A (1990), 221-243. doi: 10.1017/S0308210500031498. [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. [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), 147-151. [5] B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. [6] M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011. [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, Methods & Applications, 56 (2004), 515 - 554. doi: 10.1016/j.na.2003.09.023. [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), 693-706. [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), 252-272. doi: 10.1016/S0022-247X(03)00037-4. [10] M. Nakao and N. Aris, On global attractor for nonlinear parabolic equation of $m$-Laplacian type, J. Math. Anal. Appl., 331 (2007), 793-809. doi: 10.1016/j.jmaa.2006.08.044. [11] M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation, J. Math. Anal. Appl., 337 (2007), 1130-1142. doi: 10.1016/j.jmaa.2006.04.085. [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), 449-468. doi: 10.1619/fesi.50.449. [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), 1-17. doi: 10.1155/2009/563767. [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), 601-615. doi: 10.1016/j.jmaa.2005.05.003. [15] A. Kh. Khanmamedov, Global attractors for one dimensional $p$-Laplacian equation, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 155-171. doi: 10.1016/j.na.2008.10.037. [16] M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a $p$-Laplacian equation in $R^n$, Nonlinear Analysis: Theory, Methods and Applications, 66 (2007), 1-13. doi: 10.1016/j.na.2005.11.004. [17] C. T. Anh and T. D. Ke, Long time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 4415-4422. doi: 10.1016/j.na.2009.02.125. [18] C. T. Anh and T. D. Ke, On quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Differential Equations and Applications, 17 (2010), 195-212. doi: 10.1007/s00030-009-0048-3. [19] A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement-dependent damping, Math. Methods Appl. Sci., 33 (2010), 177-187. doi: 10.1002/mma.1161. [20] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Mathematical Surveys Monographs, 49, American Mathematical Society, 1997. [21] J. Simon, Compact sets in the space $L_p(0, T;B)$, Annali Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [22] M. A. Krasnoselskii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces," P. Noordhoff Ltd., Groningen, 1961. [23] J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," 1, Springer-Verlag, New York-Heidelberg, 1972. [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), 25- 60; Russian Math. Surveys, 42 (1987), 27-73 (English Transl.). doi: 10.1070/RM1987v042n06ABEH001503.
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