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Zbl 1132.35018
Ma, Shan; Zhong, Chengkui
The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces. (English)
[J] Discrete Contin. Dyn. Syst. 18, No. 1, 53-70 (2007). ISSN 1078-0947; ISSN 1553-5231/e

The authors introduce for weakly dissipative problems (in particular for weakly damped non-autonomous hyperbolic equation) a new class of functions, which are more general than translation compact so far used in the study of long-time behaviour of non-autonomous equations of mathematical physics. Subsequently, they study the uniform attractors for weakly damped non-autonomous hyperbolic equations with this new class (satisfying so-called $C^*$-condition) of time-dependent external forces $g(t,x)$ and prove the existence of the uniform attractors for the equation $${\partial^2 u\over\partial t^2}+\alpha{\partial u\over\partial t}- \Delta_x u+ f(u)= g(t, x),\quad u|_{\partial\Omega}= 0,$$ $$u(\tau,x)= u_\tau(x),\ \partial_t u(\tau, x)= p_\tau(x),\quad \alpha> 0,$$ where $\Omega$ is a bounded domain in $\bbfR^N$ and $f$, $g$, $u_\tau$, $p_\tau$ satisfying some natural conditions.
[Messoud A. Efendiev (Berlin)]

MSC 2000:: *35B41 Attractors
35B40 Asymptotic behavior of solutions of PDE
58J45 Hyperbolic equations
35L70 Second order nonlinear hyperbolic equations
35L20 Second order hyperbolic equations, boundary value problems

Keywords: $C^*$-condition; time-dependent external forces

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