  We study the evolution of isolated self-interacting dark matter (SIDM) halos
using spherically-symmetric gravothermal equations allowing for the scattering
cross section to be velocity dependent. We focus our attention on the large
class of models where the core is in the long mean free path regime for a
substantial time. We find that the temporal evolution exhibits an approximate
universality that allows velocity-dependent models to be mapped onto
velocity-independent models in a well-defined way using the scattering
timescale computed when the halo achieves its minimum central density. We show
how this timescale depends on the halo parameters and an average cross section
computed at the central velocity dispersion when the central density is
minimum. The predicted collapse time is fully defined by the scattering
timescale, with negligible variation due to the velocity dependence of the
cross section. We derive new self-similar solutions that provide an analytic
understanding of the numerical results.
