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**Lectures on nonlinear hyperbolic differential equations.**
*(English)*
Zbl 0881.35001

Mathématiques & Applications (Berlin). 26. Paris: Springer. vii, 289 p. (1997).

This book is a revised and extended version of widely circulated but unpublished notes from lectures given by the author at the University of Lund, during 1986-87. It concerns selected topics on nonlinear hyperbolic differential equations. Three main themes are discussed. First, the author gives theorems of existence and uniqueness of the Cauchy problem for nonlinear first-order hyperbolic systems. More precisely, after a preliminary chapter on ordinary differential operators, the Burger’s equation is studied in detail, as introduction to systems of conservation laws. These are considered in the subsequent chapters, by means of the Glimm’s method and the compensated compactness of Tartar. The material in this first part is standard, nevertheless, we think there do not exist other books collecting it in such a selfcontained and quick form.

The second part of the volume is devoted to the problem of the global existence of the solutions for nonlinear perturbations of the wave equation and Klein-Gordon equation; the material here corresponds largely to S. Klainerman [Commun. Pure Appl. Math., 33, 43-101 (1980; Zbl 0405.35056)] and subsequent papers of Klainerman, Christdoulou, and Shatah.

The conclusive chapters of the book are devoted to the paradifferential calculus, with application to propagation of singularities for fully nonlinear equations. After a preliminary chapter reviewing linear microlocal analysis, the paradifferential operators are presented in the frame of the pseudodifferential calculus of type 1,1 [cf. L. Hörmander, Commun. Partial Differ. Equ. 13, No. 9, 1085-1111 (1988; Zbl 0667.35078)]. With respect to the original exposition of J.-M. Bony [Ann. Sci. Ec. Norm. Supér., IV. Sér. 14, 209-246 (1981; Zbl 0495.35024)], here there is more generality, paid for by a loss of simplicity.

Summing up, the first part of the volume turns out to be very useful for a beginner on nonlinear hyperbolic problems, whereas the subsequent two parts, with specialistic character, are mainly addressed to experts in the field, willing to learn Fourier and microlocal techniques.

The second part of the volume is devoted to the problem of the global existence of the solutions for nonlinear perturbations of the wave equation and Klein-Gordon equation; the material here corresponds largely to S. Klainerman [Commun. Pure Appl. Math., 33, 43-101 (1980; Zbl 0405.35056)] and subsequent papers of Klainerman, Christdoulou, and Shatah.

The conclusive chapters of the book are devoted to the paradifferential calculus, with application to propagation of singularities for fully nonlinear equations. After a preliminary chapter reviewing linear microlocal analysis, the paradifferential operators are presented in the frame of the pseudodifferential calculus of type 1,1 [cf. L. Hörmander, Commun. Partial Differ. Equ. 13, No. 9, 1085-1111 (1988; Zbl 0667.35078)]. With respect to the original exposition of J.-M. Bony [Ann. Sci. Ec. Norm. Supér., IV. Sér. 14, 209-246 (1981; Zbl 0495.35024)], here there is more generality, paid for by a loss of simplicity.

Summing up, the first part of the volume turns out to be very useful for a beginner on nonlinear hyperbolic problems, whereas the subsequent two parts, with specialistic character, are mainly addressed to experts in the field, willing to learn Fourier and microlocal techniques.

Reviewer: L.Rodino (Torino)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35L40 | First-order hyperbolic systems |

35L60 | First-order nonlinear hyperbolic equations |

35S50 | Paradifferential operators as generalizations of partial differential operators in context of PDEs |

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |

35A20 | Analyticity in context of PDEs |