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I am interested in studying the basics of PDEs. I cannot find a good book on PDEs. I want a book that is more on the abstract side and explains why (so not Evan's). For example, if the topic is second order linear PDEs, I want the book to explain why they are classified into elliptic, hyperbolic and parabolic, and then explain how the time coordinate can be separated in the hyperbolic and parabolic cases (preferably before trying to solve them). I have a fairly strong background in introductory mathematics and prefer that the book takes advantage of functional analysis (but with motivation) and (sometimes) considers general Banach spaces. I want specific PDEs (such as Laplace's equation) to only be examples.

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The book by Renardy and Rogers does a much better job of explaining this kind of thing than Evans (of course, others might have a different opinion).

https://www.amazon.com/gp/aw/d/1441918205/ref=mp_s_a_1_1?ie=UTF8&qid=1486424647&sr=8-1&pi=AC_SX236_SY340_FMwebp_QL65&keywords=renardy+rogers+partial&dpPl=1&dpID=41qkS5uoxSL&ref=plSrch

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