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.
Best Answer
I have to kindly dissent from Deane Yang's recommendation of the books that I coauthored. The reason being that the question by The Common Crane is about basic references for parabolic PDE and he/she is interested in Kaehler--Ricci flow, where many cases can be reduced to a single complex Monge-Ampere equation, and hence the nature of techniques is quite different than that for Riemannian Ricci flow.
I would second user23078's recommendation of Avner Friedman's Partial Differential Equations of Parabolic Type (beautifully and carefully written), and Andras Batkai's recommendation of Nick Krylov's book Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, AMS Graduate Studies in Mathematics Volume 96 [11-18-2013: this is a correction: earlier I quoted the wrong book [Hoelder spaces]] (my colleague Lei Ni will use this book next quarter for the second part of the graduate PDE course at UCSD). To the books mentioned in the above posts on elliptic and general PDE, I would add the book by Qing Han and Fanghua Lin titled Elliptic Partial Differential Equations (in my opinion, it is a really good book; I've been using it in the PDE graduate class this quarter).
For many aspects of Ricci flow, perhaps the same may be said for Kaehler--Ricci flow, one does not need to know much parabolic PDE per se (although knowing more always helps). A fair percentage of estimates in Ricci flow are what one could call Bochner formulas. In this sense, it may be helpful to see some classic examples. A few that come to mind (perhaps not representative), are:
(1) The Bochner formula for 1-forms on a closed Riemannian manifold, which shows that there are no harmonic 1-forms if $\operatorname{Ric}>0$.
(2) The Weyl estimate used in the Weyl embedding problem; see e.g. Nirenberg's paper.
(3) Lichnerowicz formula for estimating $\lambda_1$ when $\operatorname{Ric}$ is bounded from below by a positive constant.
(4) For modern geometric analysis, the works of S.T. Yau and his coauthors on the many applications of PDE (especially the maximum principle) to problems in geometry. For parabolic equations, his paper with Peter Li on differential Harnack estimates for heat-type equations is an absolute must read. Anecdotally, I was once in Nick Krylov's house in Minnesota and I wandered into his basement and I went into his study. There, sitting prominently on the center of his table, was Li and Yau's paper. I thus learned a trade secret.
(5) Regarding Kaehler--Ricci flow, I would look into the specific techniques and calculations that workers in the field use and perform. For some of the references they quote, they may just use the techniques; for other references that they quote, they may just use the results. It may be more efficient to focus on those aspects that are most pertinent to what you are studying.
The above comments only refer to a minuscule, albeit fundamental, part of the landscape.