# American Institute of Mathematical Sciences

January  2007, 18(1): 53-70. doi: 10.3934/dcds.2007.18.53

## The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, China, China

Received  March 2006 Revised  October 2006 Published  February 2007

For weakly damped non-autonomous hyperbolic equations, we introduce a new concept Condition (C*), denote the set of all functions satisfying Condition (C*) by L2 C* $(R;X)$ which are translation bounded but not translation compact in $L^2$ loc$(R;X)$, and show that there are many functions satisfying Condition (C*); then we study the uniform attractors for weakly damped non-autonomous hyperbolic equations with this new class of time dependent external forces $g(x,t)\in$ L2 C* $(R;X)$ and prove the existence of the uniform attractors for the family of processes corresponding to the equation in $H^1_0\times L^2$ and $D(A)\times H^1_0$.
Citation: Shan Ma, Chengkui Zhong. The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 53-70. doi: 10.3934/dcds.2007.18.53
 [1] Zhenghuan Gao, Peihe Wang. Global $C^2$-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1201-1223. doi: 10.3934/dcds.2021152 [2] Sergey Zelik. Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 781-810. doi: 10.3934/dcdsb.2015.20.781 [3] Lan Wen. A uniform $C^1$ connecting lemma. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 257-265. doi: 10.3934/dcds.2002.8.257 [4] Thierry Champion, Luigi De Pascale. On the twist condition and $c$-monotone transport plans. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1339-1353. doi: 10.3934/dcds.2014.34.1339 [5] Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305 [6] Salvador Addas-Zanata, Fábio A. Tal. Support of maximizing measures for typical $\mathcal{C}^0$ dynamics on compact manifolds. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 795-804. doi: 10.3934/dcds.2010.26.795 [7] Snir Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. Journal of Modern Dynamics, 2018, 13: 43-113. doi: 10.3934/jmd.2018013 [8] Doris Bohnet. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. Journal of Modern Dynamics, 2013, 7 (4) : 565-604. doi: 10.3934/jmd.2013.7.565 [9] Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 [10] Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative lattice dynamical systems with delays. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 643-663. doi: 10.3934/dcds.2008.21.643 [11] 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 [12] Ming Mei, Yau Shu Wong, Liping Liu. Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (I) Existence and uniform boundedness. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 825-837. doi: 10.3934/dcdsb.2007.7.825 [13] José A. Conejero, Alfredo Peris. Hypercyclic translation $C_0$-semigroups on complex sectors. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1195-1208. doi: 10.3934/dcds.2009.25.1195 [14] Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361 [15] Jérôme Bertrand. Prescription of Gauss curvature on compact hyperbolic orbifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1269-1284. doi: 10.3934/dcds.2014.34.1269 [16] Andrey Gogolev. Partially hyperbolic diffeomorphisms with compact center foliations. Journal of Modern Dynamics, 2011, 5 (4) : 747-769. doi: 10.3934/jmd.2011.5.747 [17] Yazhou Han. Integral equations on compact CR manifolds. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2187-2204. doi: 10.3934/dcds.2020358 [18] Martin Michálek, Dalibor Pražák, Jakub Slavík. Semilinear damped wave equation in locally uniform spaces. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1673-1695. doi: 10.3934/cpaa.2017080 [19] Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343 [20] César Augusto Bortot, Wellington José Corrêa, Ryuichi Fukuoka, Thales Maier Souza. Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1367-1386. doi: 10.3934/cpaa.2020067

2021 Impact Factor: 1.588

## Metrics

• PDF downloads (100)
• HTML views (0)
• Cited by (10)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]