This is probably not what you are looking for, but the book An Introduction to Ordinary Differential Equations by Earl A. Coddington considers existence and uniqueness of classical solutions to $Pu=0$ (given initial conditions). Maybe the uniqueness part can be altered to include weak solutions as well.

Edit (since I don't have enough points to comment, I'll put my response here): Yes I am aware that it is about regularity, but since we have existence of a classical solution to this problem, uniqueness is enough, or have I missed something?