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Lectures on nonlinear wave equations. (English) Zbl 1089.35500
Monographs in Analysis 2. Boston, MA: International Press (ISBN 1-57146-032-2). vi, 159 p. (1995).
This welcome book starts by providing a self-contained account of the basic facts concerning the linear wave equation and the methods from harmonic analysis that are necessary when studying properties of solutions. It then proceeds to an examination of quasilinear equations with small initial data, where the Klainerman-Sobolev inequalities are introduced and global existence in higher dimensions is considered.
Leading on from this, semilinear equations with small initial data are then examined and John’s existence theorem for \(R^{1+3}\) is discussed, together with blow-up problems and some results for spherically symmetric problems. The global existence problem for semilinear equations with large data is then addressed and, after discussing the main existence results for \(R^{1+3}\), energy estimates in the subcritical case are given along with a decay lemma in the critical case. The book is to be recommended to anyone who wishes to gain a sound insight into the theoretical background necessary when studying nonlinear wave equations.

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations