Necessary condition on the dissipative term for the asymptotically free property of dissipative Kirchhoff equations

It is known that the solutions of linear abstract dissipative wave equation with integrable coefficient of the dissipation term are asymptotically free, namely, the solutions have the same asymptotic behavior as that of the solutions of an abstract free wave equation.In this paper, we conclude that the integrability is not enough for asymptotic free property of Kirchhoff equation by showing the following. For the existence of a non-trivial asymptotically free solution of abstract Kirchhoff equation with initial data in some class, an essentially stronger decay condition on the coefficient of dissipation term is necessary.