2011, 31(1): 119-138. doi: 10.3934/dcds.2011.31.119

Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent

1. 

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

Received  March 2010 Revised  December 2010 Published  June 2011

In this paper the long time behaviour of the solutions of the 3-D strongly damped wave equation is studied. It is shown that the semigroup generated by this equation possesses a global attractor in $H_{0}^{1}(\Omega )\times L_{2}(\Omega )$ and then it is proved that this is also a global attractor in $(H^{2}(\Omega )\cap H_{0}^{1}(\Omega ))\times H_{0}^{1}(\Omega )$.
Citation: A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119
References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992).

[2]

A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities,, Pacific J. Math., 207 (2002), 287.

[3]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, \arXiv{1010.4991}., ().

[4]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Memoirs of AMS, 195 (2008).

[5]

M. Conti and V. Pata, On the regulariaty of global attractors,, Discrete Contin. Dynam. Systems, 25 (2009), 1209.

[6]

B. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process,, SIAM J. Math. Anal., 33 (2002), 1090.

[7]

V. Kalantarov, Attractors for some nonlinear problems of mathematical physics,, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 50.

[8]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, J. Diff. Equations, 247 (2009), 1120.

[9]

A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement-dependent damping,, Math. Methods Appl. Sci., 33 (2010), 177.

[10]

A. Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping,, Nonlinear Analysis, 72 (2010), 1993.

[11]

A. Kh. Khanmamedov, A strong global attractor for 3-D wave equations with displacement dependent damping,, Appl. Math. Letters, 23 (2010), 928.

[12]

J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications,", \textbf{1}, 1 (1972).

[13]

W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory,, SIAM J. Appl. Math., 46 (1986), 171.

[14]

V. Pata and M. Squassina, On the strongly damped wave equation,, Commun. Math. Phys., 253 (2005), 511.

[15]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495.

[16]

V. Pata and S. Zelik, Global and exponential attractors for 3-D wave equations with displacement dependent damping,, Math. Methods Appl. Sci., 29 (2006), 1291.

[17]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (1988).

[18]

J. Simon, Compact sets in the space $L_p(0, T;B)$,, Annali Mat. Pura Appl., 146 (1987), 65.

[19]

C. Sun, D. Cao and J. Duan, Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743.

[20]

M. Yang and C. Sun, Attractors for strongly damped wave equations,, Nonlinear Analysis: Real World Applications, 10 (2009), 1097.

[21]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal., 3 (2004), 921.

[22]

S. Zhou, Global attractor for strongly damped nonlinear wave equations,, Funct. Diff. Eqns., 6 (1999), 451.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992).

[2]

A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities,, Pacific J. Math., 207 (2002), 287.

[3]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, \arXiv{1010.4991}., ().

[4]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Memoirs of AMS, 195 (2008).

[5]

M. Conti and V. Pata, On the regulariaty of global attractors,, Discrete Contin. Dynam. Systems, 25 (2009), 1209.

[6]

B. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process,, SIAM J. Math. Anal., 33 (2002), 1090.

[7]

V. Kalantarov, Attractors for some nonlinear problems of mathematical physics,, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 50.

[8]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, J. Diff. Equations, 247 (2009), 1120.

[9]

A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement-dependent damping,, Math. Methods Appl. Sci., 33 (2010), 177.

[10]

A. Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping,, Nonlinear Analysis, 72 (2010), 1993.

[11]

A. Kh. Khanmamedov, A strong global attractor for 3-D wave equations with displacement dependent damping,, Appl. Math. Letters, 23 (2010), 928.

[12]

J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications,", \textbf{1}, 1 (1972).

[13]

W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory,, SIAM J. Appl. Math., 46 (1986), 171.

[14]

V. Pata and M. Squassina, On the strongly damped wave equation,, Commun. Math. Phys., 253 (2005), 511.

[15]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495.

[16]

V. Pata and S. Zelik, Global and exponential attractors for 3-D wave equations with displacement dependent damping,, Math. Methods Appl. Sci., 29 (2006), 1291.

[17]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (1988).

[18]

J. Simon, Compact sets in the space $L_p(0, T;B)$,, Annali Mat. Pura Appl., 146 (1987), 65.

[19]

C. Sun, D. Cao and J. Duan, Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743.

[20]

M. Yang and C. Sun, Attractors for strongly damped wave equations,, Nonlinear Analysis: Real World Applications, 10 (2009), 1097.

[21]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal., 3 (2004), 921.

[22]

S. Zhou, Global attractor for strongly damped nonlinear wave equations,, Funct. Diff. Eqns., 6 (1999), 451.

[1]

Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015

[2]

Veronica Belleri, Vittorino Pata. Attractors for semilinear strongly damped wave equations on $\mathbb R^3$. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 719-735. doi: 10.3934/dcds.2001.7.719

[3]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1/2) : 31-52. doi: 10.3934/dcds.2004.10.31

[4]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

[5]

Alexandre N. Carvalho, Jan W. Cholewa. Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$. Conference Publications, 2007, 2007 (Special) : 230-239. doi: 10.3934/proc.2007.2007.230

[6]

Fabrizio Colombo, Davide Guidetti. Identification of the memory kernel in the strongly damped wave equation by a flux condition. Communications on Pure & Applied Analysis, 2009, 8 (2) : 601-620. doi: 10.3934/cpaa.2009.8.601

[7]

Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094

[8]

Piotr Kokocki. Homotopy invariants methods in the global dynamics of strongly damped wave equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3227-3250. doi: 10.3934/dcds.2016.36.3227

[9]

Björn Birnir, Kenneth Nelson. The existence of smooth attractors of damped and driven nonlinear wave equations with critical exponent , s = 5. Conference Publications, 1998, 1998 (Special) : 100-117. doi: 10.3934/proc.1998.1998.100

[10]

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

[11]

Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357

[12]

Pierre Fabrie, Cedric Galusinski, A. Miranville, Sergey Zelik. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1/2) : 211-238. doi: 10.3934/dcds.2004.10.211

[13]

Lili Fan, Hongxia Liu, Huijiang Zhao, Qingyang Zou. Global stability of stationary waves for damped wave equations. Kinetic & Related Models, 2013, 6 (4) : 729-760. doi: 10.3934/krm.2013.6.729

[14]

P. Fabrie, C. Galusinski, A. Miranville. Uniform inertial sets for damped wave equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 393-418. doi: 10.3934/dcds.2000.6.393

[15]

Alexandre Nolasco de Carvalho, Jan W. Cholewa, Tomasz Dlotko. Damped wave equations with fast growing dissipative nonlinearities. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1147-1165. doi: 10.3934/dcds.2009.24.1147

[16]

Bixiang Wang, Xiaoling Gao. Random attractors for wave equations on unbounded domains. Conference Publications, 2009, 2009 (Special) : 800-809. doi: 10.3934/proc.2009.2009.800

[17]

Kyouhei Wakasa. The lifespan of solutions to semilinear damped wave equations in one space dimension. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1265-1283. doi: 10.3934/cpaa.2016.15.1265

[18]

Petronela Radu, Grozdena Todorova, Borislav Yordanov. Higher order energy decay rates for damped wave equations with variable coefficients. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 609-629. doi: 10.3934/dcdss.2009.2.609

[19]

Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2/3) : 351-392. doi: 10.3934/dcds.2004.11.351

[20]

Shengfan Zhou, Linshan Wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 399-412. doi: 10.3934/dcds.2003.9.399

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]