A discussion of the vacuum Einstein equations (only gravity, no other fields) in terms of synthetic differential geometry is in

Gonzalo Reyes, A derivation of Einstein’s vacuum field equations (pdf)

PDE theory

Genuine PDE theory for Einstein’s equations goes back to local existence results by Yvonne Choquet-Bruhat in the 1950s. Global existence in the presence of a Cauchy surface was then shown in

Sergiu Klainerman, Francesco Nicolo, The evolution problem in general relativity, Progress in Mathematical Physics, 25. Birkhäuser Boston, Inc., Boston, MA, 2003. xiv+385 pp. ISBN: 0-8176-4254-4

For further developments see

H. Friedrich, A. D. Rendall, The Cauchy Problem for the Einstein Equations (arXiv:gr-qc/0002074)

Alan D. Rendall, Partial differential equations in general relativity, Oxford University press 2008 (web)

Hans Ringström, The Cauchy Problem in General Relativity, ESI Lectures in Mathematics and Physics 2009 (web)