Forward dynamics of strongly wave equations with asymptotically vanishing damping on $ {\mathbb{R}}^3 $

This work is devoted to the forward asymptotic behavior of solutions to a class of non-autonomous strongly damped wave equations on $ {\mathbb{R}}^3 $. The main feature of the equation is that the damping effect is allowed to vanish as time goes to infinity. This makes the standard locally uniformly boundedness of the force insufficient to ensure the ultimate boundedness of solutions, and as a consequence the usual uniform attractor theory does not apply here. In this paper, by introducing a new integral condition of the force in terms of the vanishing order of the damping term, we shall prove the existence of a compact minimal forward attractor as an alternative of the standard uniform attractor.