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

show all references

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

[1]

Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation. Communications on Pure & Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023

[2]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

[3]

T. F. Ma, M. L. Pelicer. Attractors for weakly damped beam equations with $p$-Laplacian. Conference Publications, 2013, 2013 (special) : 525-534. doi: 10.3934/proc.2013.2013.525

[4]

Jacson Simsen, José Valero. Global attractors for $p$-Laplacian differential inclusions in unbounded domains. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3239-3267. doi: 10.3934/dcdsb.2016096

[5]

Kerstin Does. An evolution equation involving the normalized $P$-Laplacian. Communications on Pure & Applied Analysis, 2011, 10 (1) : 361-396. doi: 10.3934/cpaa.2011.10.361

[6]

Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033

[7]

Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou. Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 405-440. doi: 10.3934/dcds.2009.24.405

[8]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107

[9]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361

[10]

Pavel Jirásek. On Compactness Conditions for the $p$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 715-726. doi: 10.3934/cpaa.2016.15.715

[11]

Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371

[12]

Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731

[13]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315

[14]

Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683

[15]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070

[16]

Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22

[17]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

[18]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058

[19]

Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026

[20]

Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 555-567. doi: 10.3934/dcdsb.2019254

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]