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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

Shan Ma 1, and  Chengkui Zhong 1,

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 $D(A)\times H^1_0$.
Keywords: uniform attractor; translation compact; uniformly ω-limit compact; uniform Condition (C); Condition (C*); damped hyperbolic equations..
Mathematics Subject Classification: Primary: 35B40, 35B41, 58J4.
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
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