  We investigate the long time behavior of solutions to semilinear hyperbolic
equation (E$_{\alpha}$): $
u^{\prime\prime}(t)+\gamma(t)u^{\prime}(t)+Au(t)+f(u(t))=g(t),~t\geq0, $ where
$A$ is a self-adjoint nonnegative operator, $f$ a function which derives from a
convex function, and $\gamma$ a nonnegative function which behaviors, for $t$
large enough, as $\frac{K}{t^{\alpha}}$ with $K>0$ and $\alpha \in\lbrack0,1[.$
We obtain sufficient conditions on the source term $g(t),$ ensuring the weak or
the strong convergence of any solution $u(t)$ of (E$_{\alpha}$) as
$t\rightarrow+\infty$ to a solution of the stationary equation $Av+f(v)=0$ if
one exists.
