Asymptotic limits for mildly degenerate Kirchhoff equations

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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.