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Blow-up in quasilinear parabolic equations. Transl. from the Russian by Michael Grinfeld. (English) Zbl 1020.35001
de Gruyter Expositions in Mathematics. 19. Berlin: Walter de Gruyter. xxi, 533 p. (1995).
From the preface: In the relatively brief time that has passed since the appearance of this book in [Peaking modes in problems for quasilinear parabolic equations (Russian), Nauka, Moscow (1987; Zbl 0631.35002)], a range of new results have been obtained in the theory of strongly nonstationary evolution equations, the main problems of this area have been more clearly delineated, specialist monographs and a large number of research papers were published, and the sphere of applications has expanded. It turns out that, as far as nonlinear heat equations with a source term are concerned, the present authors have, on the whole, correctly indicated the main directions of development of the theory of finite-time blow-up processes in nonlinear media. We were gratified to see that the subject matter of the book has lost none of its topicality; in fact, its implications have widened. Therefore we thought it right to confine ourselves to relatively insignificant additions and corrections in the body of the work.
In preparing the English edition we have included additional material, provided an updated list of references and reworked the comments sections whenever necessary. It is well known that most phenomena were discovered by analyzing simple particular solutions of the equations and systems under consideration. This also applies to the theory of finite-time blow-up. We include in the introductory Chapters I and II, and in Chapter IV, new examples of unusual special solutions, which illustrate unexpected properties of unbounded solutions and pose open problems concerning asymptotic behaviour. Some of these solutions are not self-similar (or invariant with respect to a group of transformations). Starting from one such solution and using the theory of intersection comparison of unbounded solutions having the same existence time, we are able to obtain new optimal estimates of evolution of fairly arbitrary solutions. This requires changing the manner of presentation of the main comparison results and some subsequent material in Chapter IV.

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35K55 Nonlinear parabolic equations
35B44 Blow-up in context of PDEs