  We study the asymptotic behavior of the solutions of the mildly degenerate
Kirchhoff equation with a dissipative term. We obtain a new estimate on
second-in-time derivative of the solution. Moreover we renormalize the solution
in such a way that the renormalization as a no zero limit as t goes to
infinity. Finally we calculate explicitly the norm of this limit and we prove
that it does not depend on the initial data.
