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

*(English)*Zbl 1132.35018The 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

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

$$u(\tau ,x)={u}_{\tau}\left(x\right),\phantom{\rule{4pt}{0ex}}{\partial}_{t}u(\tau ,x)={p}_{\tau}\left(x\right),\phantom{\rule{1.em}{0ex}}\alpha >0,$$

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

Reviewer: Messoud A. Efendiev (Berlin)