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

Asymptotically free property of the solutions of an abstract linear hyperbolic equation with time-dependent coefficients

“…The result of this paper is applied to dissipative Kirchhoff equations in [15] to obtain the necessary decay condition on the dissipative term for the asymptotically free property. This condition is essentially stronger than that of linear dissipative wave equation.…”