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?

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?

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.

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?