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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 53 - 70, Volume 18, Issue 1, May 2007      doi:10.3934/dcds.2007.18.53

 
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Shan Ma - School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, China (email)
Chengkui Zhong - School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, China (email)

Abstract: 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 $D(A)\times H^1_0$.

Keywords:  uniform attractor; translation compact; uniformly ω-limit compact; uniform Condition (C); Condition (C*); damped hyperbolic equations.
Mathematics Subject Classification:  Primary: 35B40, 35B41, 58J45.

Received: March 2006;      Revised: October 2006;      Available Online: February 2007.