We consider Kirchhoff equations with a small parameter $\varepsilon$ in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to $0$ as $t\to +\infty$ (weak dissipation).
In this note we present some recent results concerning existence of global solutions, and their asymptotic behavior both as $t\to +\infty$ and as $\varepsilon\to 0^{+}$. Since the limit equation is of parabolic type, this is usually referred to as a hyperbolic-parabolic singular perturbation problem.
We show in particular that the equation exhibits hyperbolic or parabolic behavior depending on the values of the parameters.
