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I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly and take turn to present a portion of it.

We just finished Regularity Theory for Elliptic PDE from Fernandez-Real and Ros-Oton (which was great), and now have to start a new one. To this end, we each propose a book and then vote for the one we're the most interested in, the suggestion with the highest exit poll is chosen. I am therefore investigating my options to choose wisely the one I will suggest and that is why I am seeking advice on this platform.

I would be interested in learning more about Parabolic Equations, which I know very little about. I asked my supervisor who gave me a few suggestions, but the books were mostly from the 70's. Hence, I was wondering whether there were more state of the art books considered as benchmark in this realm. I have heard of Elliptic and Parabolic Equations from Wu, Yin and Wang, and I was wondering how good it was. I would especially like a book highlighting the differences between Elliptic and Parabolic equations, and explaining why introducing this time dependance changes the theory that much.

Any answer and suggestion is more than welcome.

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    $\begingroup$ In a subject with such a tradition, you can do worst than starting with the study of good books from the 60s and the 70s. $\endgroup$
    – Alessandro Della Corte
    Commented Aug 12 at 14:47
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    $\begingroup$ I recommend "Elliptic and Parabolic equations in Sobolev spaces" by N. Krylov. $\endgroup$
    – Giorgio Metafune
    Commented Aug 14 at 8:16

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Premise

This is a long, possibly tedious, comment. In my opinion, the problem with new books on such topics is that in most cases their contents are either expositions of the classical theory or specialised monographs on particular topics. In the first case there's nothing new except possibly the didactical skills of the Author(s), while in the second case usually there are several interesting/important results of necessarily restricted applicability/scope.
Again in my opinion, this situation is due to the fact that after the great progress in the the general theory of PDEs and in particular on the theory of parabolic ones done in the 50' and the 60' of the twentieth century, things slowed down. For example,

  • While the uniqueness problem or the heat equation (the simplest parabolic equation) with initial data having a controlled growth respect to the spatial variable has been settled by Tikhonov, Nicolescu and Täcklind after the initial result of Mauro Picone, the uniqueness problem fo solutions having rapidly oscillating time behaviour around $t=0$ is still open, even if several partial results are known (see, in this respect, this old question by Zen Harper).
  • Even the reasons of the differences between the theories of the three basic classes of PDEs are possibly dealt with in an unsatisfactory manner. We know that the main "ingredient" which gives the solutions to a given PDE their general properties is the geometry of its characteristic (hyper-)surface, but the reason aren't, to my knowledge, described in the due details in no single source, even if there are hints in every introductory textbook on the theory of PDEs (for example Treves Basic linear partial differential operators, or the first volume of Taylor's Partial differential equations)

Advices

After the above premise (which may sound like as a "it is not possible to answer to your question"), just for the sake of completeness, I'd like to cite a further two titles in addition to the ones given by Carlo in his answer.

The classical work [1] is, to my knowledge, the most complete work on the topic even if

  1. it is from the sixties, contrarily to what the Asker required in their OP, and
  2. for the case of discontinuous coefficients, there are other, more complete references.

The other reference [2] has an entirely different flavour (and it is one of the books I like more) respect to the other

  1. even it is completely rigorous, it is application oriented and deals also with (nonlinear) integrodifferential equations with various (nonlinear) boundary / initial value conditions,
  2. the ambient function spaces are the spaces $C^{k,\alpha}$, for $k\in \Bbb N$ and $\alpha\in]0,1]$ and finally
  3. the method applied by professor Pao for the solution of the proposed problems is the method of monotone sequence ad it is based on the application of the maximum principle (which holds for elliptic and parabolic equations bu not for hyperbolic ones).

References

[1] Ol'ga Alexandrovna Ladyzhenskaya, Vsevolod Alekseevich Solonnikov, and Nina Nikolaevna Ural’tseva, Linear and quasi-linear equations of parabolic type. Translated from the Russian by S. Smith. (English) Translations of Mathematical Monographs. 23. Providence, RI: American Mathematical Society (AMS), pp. XI+648 (1968), DOI:10.1090/mmono/023, MR241822, Zbl 0174.15403.

[2] Chia-Ven Pao (1992), Nonlinear Parabolic and Elliptic Equations, Plenum Press, xv+777, DOI:10.1007/978-1-4615-3034-3, MR1212084, Zbl 0777.35001.

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    $\begingroup$ The book [1] is importantc and widely cited. The biggest problem with this book is that its notations are old and very hard to read. I don't get why AMS has not released it with modern typesetting $\endgroup$
    – Akira
    Commented Aug 17 at 16:07
  • $\begingroup$ @Akira, possibly because this would be a quite expensive operation: I am aware of only one classical book that has been entirely re-edited, that is S. L. Sobolev's Some Applications of Functional Analysis in Mathematical Physics (by AMS). And if you don't know if there are a sufficient number of mathematicians/libraries who are willing to purchase it, this would be a risky operation. $\endgroup$
    – Daniele Tampieri
    Commented Aug 17 at 16:20
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    $\begingroup$ The book is a classic and I guess most libraries (and researchers using PDE) would love to have its re-edited edition. Even just typesetting the book in LaTeX is already very beneficial very the communities. I guess just typesetting the book is not expensive at all, no? $\endgroup$
    – Akira
    Commented Aug 17 at 19:58
  • $\begingroup$ @Akira, I agree about the interest of the book, for it is always cited as an essential reference when dealing with parabolic PDEs. However, typesetting may require up to a hundred hours as it requires someone who's able to update correctly the notation, and this may require a competent and costly editor, not just a clever $\LaTeX$ user. $\endgroup$
    – Daniele Tampieri
    Commented Aug 17 at 21:52
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Back in 2012, professor Ben Chow gave some advice to a similar question; these include the

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    $\begingroup$ +1. Possibly Friedman gives an overview of many techniques used in parabolic PDE theory and he's also aware of the uniqueness problems for the Cauchy problem. $\endgroup$
    – Daniele Tampieri
    Commented Aug 16 at 9:06

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