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DCDS-B

In this paper, we are concerned with
some properties of the global attractor of weakly damped wave
equations. We get the existence of multiple stationary solutions for
wave equations with weakly damping. Furthermore, we provide some
approaches to verify the small neighborhood of the origin is an
attracting domain which is important to obtain
the multiple equilibrium points in global attractor.

keywords:
wave equations.
,
global
attractor
,
Lyapunov functional
,
$Z_2$ index
,
equilibrium points

DCDS-B

In this paper, we consider the long time behavior of the solution for the following nonlinear damped wave equation
\begin{eqnarray*}
\varepsilon(t) u_{tt}+g(u_{t})-\Delta u+\varphi (u)=f
\end{eqnarray*}
with Dirichlet boundary condition, in which, the coefficient $\varepsilon$ depends explicitly on time, the damping $g$ is nonlinear and the nonlinearity $\varphi$ has a critical growth. Spirited by this concrete problem, we establish a sufficient and necessary condition for the existence of attractors on time-dependent spaces, which is equivalent to that provided by M. Conti et al.[10]. Furthermore, we give a technical method for verifying compactness of the process
via contractive functions.
Finally, by the new framework, we obtain the existence of the
time-dependent attractors for the wave equations with nonlinear damping.

## Year of publication

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