Okay, I'm about to take an introductory course for PDE's (for some reason I got into weak solutions and such before learning the preliminaries), and so I would like to know if there are any books you would recommend that have a strong mathematical orientation (not that I don't like the many applications, but when I took an ODE's course we spent almost all semester modelling populations and such and when I read on my own Picard-Lindelöf and Peano's existence theorems, and all the mathematics behind what I was actually doing I felt cheated).