I am very intrigued by distribution theory and have already some knowledge in the subject, but I'm still looking for a source, which treats PDEs consequently with tools from this theory (as far as they reach) and relies only minimal on arguments from classical analysis. To avoid those arguments and to replace them by more elegant ones catching more of the aesthetic but often imprecise computations of physicians, seems to be one of the main motivations behind distribution theory after all.
One example: In the book of D. Mitrea: Distributions, Partial Differential Equations and Harmonic Analysis many examples of fundamental solutions are explicitly computed, but only using methods from classical analysis. For example the fundamental solution of the wave equation in 3+1 dimensions is derived by a painful 5 pages computation. As I saw in another source, the solution can also be determined very fast using the fourier transform of measures.
Do you know any book, which proceeds in this way and tries to replace classical arguments consequently by more abstract ones from distribution theory?