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Nonlinear partial differential equations with applications. (English) Zbl 1087.35002
ISNM. International Series of Numerical Mathematics 153. Basel: Birkhäuser (ISBN 3-7643-7293-1/hbk). xviii, 405 p. (2005).
The book focuses on various classes of nonlinear elliptic and parabolic partial differential equations arising in applications. Of main concern are the existence, uniqueness, regularity, and continuous dependence on data of solutions. For each class of equations, an abstract theory is first developed, followed by specific applications. The monograph starts with a preliminary chapter, where the author reviews some basic functional analytic tools.
The core of the book consists of 11 chapters that are divided into two parts: Steady-state problems (5 chapters), and evolution problems (6 chapters). Many chapters contain relevant examples and exercises, as well as useful bibliographical remarks. Some of the abstract tools reviewed in the monograph are: pseudomonotone or weakly continuous mappings, accretive operators, nonlinear semigroups, set-valued mappings, and compactness methods. Specific partial differential equations or systems of partial differential equations that are discussed include semilinear heat equations, quasilinear elliptic equations of divergence type, reaction-diffusion systems, Navier-Stokes equations, doubly nonlinear problems, and phase-field models.
The book is well written and represents a valuable addition to the literature. We recommend it to both graduate students and researchers interested in the modern theory of partial differential equations.

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations