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## Reference request: parabolic PDE

I want to learn about parabolic PDE and it seems to me that there is no established reference as far as where one should look if one wants to learn the subject from basics.

I think I have a firm grip on elliptic PDE after going through the first part of Gilbarg and Trudinger + some Monge-Ampere stuff. But that concludes my PDE background at this moment.

Can someone provide me with a good textbook for parabolic PDE? Any chunk of information will be appreciated.

EDIT: I would like to learn about parabolic PDE arising in geometry, mostly the Kahler-Ricci flow and related questions. But since I am new to this approach perhaps a more broad introduction would be appropriate.

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 Friedman's Partial Differential Equations of Parabolic Type may be a good choice. –  Shanlin Huang Oct 7 at 10:47

If you are interested in sup-norm and Hölder estimates, then Lunardi's book is a good start:

http://www.amazon.com/Semigroups-Regularity-Parabolic-Differential-Applications/dp/3764351721/ref=la_B001K6J69O_1_1?ie=UTF8&qid=1349596684&sr=1-1

Otherwise you should specify what type of equation are you interested in.

ADDED: Afret the comment of @Liviu:

You should not omit Krylov's books from your list: the one on Hölder spaces and the one on $L^p$ spaces.

And of course there is Evans. An excellent introduction.

ADDED: After the clarification in the question: Topping's lecture notes (there is also a book version from the London Mathematical Society) are quite nice and readable.

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Here are some useful references. J.L. Lions, E. Magenes: Non homogeneous boundary value problems, vol.II Springer Verlag, 1972 and A. Friedman: Partial Differential Equations, Dover 2008, H. Brezis: Functional analysis, Sobolev Spaces, and Partail differential Equations, Springer Verlag 2011. Brezis' book would be the place to start. Lions and Magenes is the most sophisticated but it only deals with Hilbert spaces, i.e., $L^2$-based Sobolev spaces. – Liviu Nicolaescu Oct 7 at 12:45
Thanks for the suggestions. As it was rightfully asked, I edited my question to reflect my motivation of study. Perhaps this narrows a few things down. Does it? – The Common Crane Oct 7 at 20:53
Somebody knowledgeable recommended Lieberman's book because it is similarly written to Gilbarg & Trudinger. Is it worth it compared to other choices? – The Common Crane Oct 10 at 0:20

If it's the Ricci flow you're really interested in, I recommend checking out the books on the Ricci flow written by Bennett Chow and his co-authors.

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