# American Institute of Mathematical Sciences

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

## Uniform exponential attractors for a singularly perturbed damped wave equation

 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.
Citation: 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
 [1] 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 [2] 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 [3] Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041 [4] Ciprian G. Gal. Robust exponential attractors for a conserved Cahn-Hilliard model with singularly perturbed boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (4) : 819-836. doi: 10.3934/cpaa.2008.7.819 [5] 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 [6] Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2019101 [7] 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 [8] 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 [9] Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823 [10] Feng Zhou, Chunyou Sun, Xin Li. Dynamics for the damped wave equations on time-dependent domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1645-1674. doi: 10.3934/dcdsb.2018068 [11] Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257 [12] Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571 [13] Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079 [14] Minbo Yang, Yanheng Ding. Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Communications on Pure & Applied Analysis, 2013, 12 (2) : 771-783. doi: 10.3934/cpaa.2013.12.771 [15] Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1001-1030. doi: 10.3934/dcds.2011.29.1001 [16] 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 [17] 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 [18] Fengjuan Meng, Meihua Yang, Chengkui Zhong. Attractors for wave equations with nonlinear damping on time-dependent space. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 205-225. doi: 10.3934/dcdsb.2016.21.205 [19] Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985 [20] 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

2017 Impact Factor: 1.179