Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution theory since a good connection exists). I have seen that the applications and/or research topics are varied. I also tried to see more about mathematical analysis, but every time I am more passionate about these topics.
I also mention this thread for applications: Applications of microlocal analysis?
For example, even if the question I'm doing is quite generic, these theories that are mostly applied to the study of linear PDEs, as can be applied in non-linear PDE contexts (I read that they are very important for this case too very difficult). The non-linear case techniques on what are based? Clearly I refer for example to the existence and uniqueness of the solution or regularity.
Another thing to add, I have read of applications in various classification of PDEs, ellipticals obviously (fairly well known), but also hyperbolic or parabolic. I would be grateful for every answer.
Clearly any reference is welcome