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The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces. (English) Zbl 1132.35018

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 $g\left(t,x\right)$ and prove the existence of the uniform attractors for the equation

$\frac{{\partial }^{2}u}{\partial {t}^{2}}+\alpha \frac{\partial u}{\partial t}-{{\Delta }}_{x}{u+f\left(u\right)=g\left(t,x\right),\phantom{\rule{1.em}{0ex}}u|}_{\partial {\Omega }}=0,$
$u\left(\tau ,x\right)={u}_{\tau }\left(x\right),\phantom{\rule{4pt}{0ex}}{\partial }_{t}u\left(\tau ,x\right)={p}_{\tau }\left(x\right),\phantom{\rule{1.em}{0ex}}\alpha >0,$

where ${\Omega }$ is a bounded domain in ${ℝ}^{N}$ and $f$, $g$, ${u}_{\tau }$, ${p}_{\tau }$ satisfying some natural conditions.

##### MSC:
 35B41 Attractors (PDE) 35B40 Asymptotic behavior of solutions of PDE 58J45 Hyperbolic partial differential equations on manifolds 35L70 Nonlinear second-order hyperbolic equations 35L20 Second order hyperbolic equations, boundary value problems