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**Partial differential equations of parabolic type.**
*(English)*
Zbl 0144.34903

Englewood Cliffs, N.J.: Prentice-Hall, Inc. xiv, 347 p. (1964).

This monograph is devoted to a rather special topic in partial differential equations, the theory of parabolic equations, and – to some extent – the parallel theory of elliptic equations. The emphasis is on the second order (classical) case, so that most of the material is quite traditional. There are 10 chapters.

In Chapter 1 the fundamental solution for a second order parabolic operator is constructed by the parametrix method of Levi. As an application the Cauchy (initial value) problem, existence and uniqueness, is considered. In Chapter 2 the maximum principle is treated, along with some of its consequences. In Chapter 3 an existence proof is given for the first initial-boundary value problem, which is based on the continuity method and the Schauder estimates. The derivation of the latter is given in Chapter 4. In Chapter 5 the second initial boundary value problem is solved by the integral equation method. Chapter 6 is concerned with the asymptotic behavior (as \(t \to\infty)\) of solutions. It contains also some results on uniqueness for the “backward” equation. Chapter 7 deals with semilinear equations and nonlinear boundary condition. In Chapter 8 free boundary problems (the so-called Stefan problem) are discussed.

Finally, the two last chapters of the monograph consider the extension to parabolic equations and systems (in the sense of Petrovskii) of any order. In Chapter 9 the fundamental solution is constructed in a similar way as in Chapter 1. In Chapter 10 is developed the Hilbert space approach to the Dirichlet problem for elliptic systems and the first initial-boundary value problem for parabolic equations. There is also an appendix, accompanied by a short separate bibliography, where the theory of second order parabolic equations with discontinuous coefficients (a, la de Giorgi-Nash-Moser) is briefly outlined. The main bibliography, preceded by three pages of “bibliographical notes”, contains 121 items.

The exposition is clear throughout but maybe sometimes too pedantic to satisfy all tastes. In our opinion, the monograph is an important addition to the existing literature on partial differential equations. It will certainly be of great value not only to the student who seriously wants to penetrate the more technical aspects of the subject but also to the specialist who may use it as a source of reference.

In Chapter 1 the fundamental solution for a second order parabolic operator is constructed by the parametrix method of Levi. As an application the Cauchy (initial value) problem, existence and uniqueness, is considered. In Chapter 2 the maximum principle is treated, along with some of its consequences. In Chapter 3 an existence proof is given for the first initial-boundary value problem, which is based on the continuity method and the Schauder estimates. The derivation of the latter is given in Chapter 4. In Chapter 5 the second initial boundary value problem is solved by the integral equation method. Chapter 6 is concerned with the asymptotic behavior (as \(t \to\infty)\) of solutions. It contains also some results on uniqueness for the “backward” equation. Chapter 7 deals with semilinear equations and nonlinear boundary condition. In Chapter 8 free boundary problems (the so-called Stefan problem) are discussed.

Finally, the two last chapters of the monograph consider the extension to parabolic equations and systems (in the sense of Petrovskii) of any order. In Chapter 9 the fundamental solution is constructed in a similar way as in Chapter 1. In Chapter 10 is developed the Hilbert space approach to the Dirichlet problem for elliptic systems and the first initial-boundary value problem for parabolic equations. There is also an appendix, accompanied by a short separate bibliography, where the theory of second order parabolic equations with discontinuous coefficients (a, la de Giorgi-Nash-Moser) is briefly outlined. The main bibliography, preceded by three pages of “bibliographical notes”, contains 121 items.

The exposition is clear throughout but maybe sometimes too pedantic to satisfy all tastes. In our opinion, the monograph is an important addition to the existing literature on partial differential equations. It will certainly be of great value not only to the student who seriously wants to penetrate the more technical aspects of the subject but also to the specialist who may use it as a source of reference.

Reviewer: Jaak Peetre (Lund)