A useful reference would be the paper
Mantegazza, Carlo, Mennucci, Andrea Carlo: Hamilton-Jacobi equations and distance functions on Riemannian manifolds.Appl. Math. Optim.47(2003), no.1, 1–25.
https://mathscinet.ams.org/mathscinet-getitem?mr=1941909
See in particular Theorem 3.1.