Kination, meet Kasner: on the asymptotic cosmology of string compactifications

We study runaway, kination-dominated epochs in string cosmology. We show how the apparent classical decompactification runaway of the volume modulus, described by a kination epoch in the 4-dimensional EFT, can be uplifted to a classical Kasner solution in 10d in which the non-compact dimensions collapse towards a Big Crunch. This can also be generalised for arbitrary spacetime and compactification dimensions. We conclude with some comments on how this picture is modified by quantum effects, and the need for both dynamical and kinematical Swampland constraints.