Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates

We consider the second order Cauchy problem for the damped Kirchhoff equation depending an a small parameter and the first order limit problem. We prove decay estimates for solutions of the first order problem, and we show that analogous estimates hold true for solutions of the second order problem provided that the parameter is small enough. We also show that our decay rates are optimal in many cases.