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

Shan Ma; Chengkui Zhong

doi:10.3934/dcds.2007.18.53

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2007, Volume 18, Issue 1: 53-70. Doi: 10.3934/dcds.2007.18.53

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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

Received: February 28, 2006

Revised: September 30, 2006

Published: December 2006

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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.

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