American Institute of Mathematical Sciences

Advanced
February  2004, 10(1&2): 211-238. doi: 10.3934/dcds.2004.10.211

Uniform exponential attractors for a singularly perturbed damped wave equation

Pierre Fabrie 1, , Cedric Galusinski 1, , A. Miranville 2, and  Sergey Zelik 3,

1. 

Université Bordeaux-I, Mathématiques Appliquées, 351 Cours de la Libération, 33405 Talence Cedex, France, France
2. 

Laboratoire d'Applications des Mathématiques - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, Chasseneuil Futuroscope Cedex, France
3. 

Université de Poitiers, Laboratoire d'Applications des Mathématiques - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex, France

Received  November 2001 Revised  March 2003 Published  October 2003

Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constructions obtained for parabolic systems cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions between the perturbed and unperturbed equations. We note that the continuity is obtained without time shifts that have been used in previous results.
Keywords: reaction-diffusion equations, time boundary layer., Singularly perturbed damped wave equations, uniform exponential attractors.
Mathematics Subject Classification: 35B40, 35B45, 35L3.
Citation: Pierre Fabrie, Cedric Galusinski, A. Miranville, Sergey Zelik. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 211-238. doi: 10.3934/dcds.2004.10.211
[1]

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

Dandan Li. Asymptotics of singularly perturbed damped wave equations with super-cubic exponent. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 583-600. doi: 10.3934/dcdsb.2021056
[3]

Valentin Butuzov, Nikolay Nefedov, Oleh Omel'chenko, Lutz Recke. Boundary layer solutions to singularly perturbed quasilinear systems. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021226
[4]

Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041
[5]

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

Ciprian G. Gal. Robust exponential attractors for a conserved Cahn-Hilliard model with singularly perturbed boundary conditions. Communications on Pure and Applied Analysis, 2008, 7 (4) : 819-836. doi: 10.3934/cpaa.2008.7.819
[7]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31
[8]

Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5673-5694. doi: 10.3934/dcdsb.2019101
[9]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations and Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039
[10]

Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015
[11]

Veronica Belleri, Vittorino Pata. Attractors for semilinear strongly damped wave equations on $\mathbb R^3$. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 719-735. doi: 10.3934/dcds.2001.7.719
[12]

Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823
[13]

Ahmed Bonfoh, Ibrahim A. Suleman. Robust exponential attractors for singularly perturbed conserved phase-field systems with no growth assumption on the nonlinear term. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3655-3682. doi: 10.3934/cpaa.2021125
[14]

Feng Zhou, Chunyou Sun, Xin Li. Dynamics for the damped wave equations on time-dependent domains. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1645-1674. doi: 10.3934/dcdsb.2018068
[15]

Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257
[16]

Jihoon Lee, Vu Manh Toi. Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3135-3152. doi: 10.3934/dcdsb.2020054
[17]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
[18]

Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079
[19]

Xiangming Zhu, Chengkui Zhong. Uniform attractors for nonautonomous reaction-diffusion equations with the nonlinearity in a larger symbol space. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3933-3945. doi: 10.3934/dcdsb.2021212
[20]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (100)
  • HTML views (0)
  • Cited by (79)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy