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

Abstract: 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.