Asymptotic for a semilinear hyperbolic equation with asymptotically vanishing damping term, convex potential, and integrable source

We investigate the long time behavior of solutions to semilinear hyperbolic equation (E${}_{\alpha }$): $u^{\mathrm{\prime }\mathrm{\prime }}(t)+\gamma (t)u^{\mathrm{\prime }}(t)+Au(t)+f(u(t))=g(t),t\ge 0,$ 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 $\alpha \in [0,1[\mathrm{.}$ 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 $t\to +\mathrm{\infty }$ to a solution of the stationary equation $Av+f(v)=0$ if one exists.