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**The Cauchy problem for partial differential operators. Paper from the 27th Brazilian Mathematics Colloquium – 27\(°\) Colóquio Brasileiro de Matemática, Rio de Janeiro, Brazil, July 27–31, 2009.
(Problema de Cauchy para operadores diferenciais parciais.)**
*(Portuguese)*
Zbl 1183.35001

Publicações Matemáticas do IMPA. Rio de Janeiro: Instituto Nacional de Matemática Pura e Aplicada (IMPA) (ISBN 978-85-244-0301-9/pbk). ii, 155 p. (2009).

As the title indicates, this work presents an analysis of the Cauchy problem for partial differential operators.

The book is divided into four chapters. In the first chapter, auxiliary concepts and results are introduced. This includes the Cauchy-Kowalewska, Holmgren and Lax-Mizohata theorems. The second chapter is devoted to nonlinear Cauchy problems. It starts with a section where nonlinear hyperbolic systems are considered. Then, the well-posedness of the Cauchy problem is proved within a class of nonlinear symmetric systems. Chapter three considers the Cauchy problem for weakly hyperbolic systems. This chapter is mainly based on the Ph.D. Thesis of the first author and on his paper [Osaka J. Math. 42, No. 4, 831–860 (2005; Zbl 1122.35066)]. In the last section of this chapter, a result due to T. Nishitani and S. Spagnolo [Osaka J. Math. 41, No. 1, 145–157 (2004; Zbl 1065.35203)] is also discussed. The results of the last chapter are based on [M. R. Ebert, R. A. Kapp and J. R. Dos Santos Filho, J. Math. Anal. Appl. 359, No. 1, 181–196 (2009; Zbl 1178.35260)]. Bounds for the optimal loss of regularity in the Sobolev scale for the Cauchy problem under consideration are presented here. To conclude, in the last section of chapter four (and the book), the authors present some remarks about this chapter and describe open problems related with well-posed Cauchy problems for weakly hyperbolic operators.

The book is divided into four chapters. In the first chapter, auxiliary concepts and results are introduced. This includes the Cauchy-Kowalewska, Holmgren and Lax-Mizohata theorems. The second chapter is devoted to nonlinear Cauchy problems. It starts with a section where nonlinear hyperbolic systems are considered. Then, the well-posedness of the Cauchy problem is proved within a class of nonlinear symmetric systems. Chapter three considers the Cauchy problem for weakly hyperbolic systems. This chapter is mainly based on the Ph.D. Thesis of the first author and on his paper [Osaka J. Math. 42, No. 4, 831–860 (2005; Zbl 1122.35066)]. In the last section of this chapter, a result due to T. Nishitani and S. Spagnolo [Osaka J. Math. 41, No. 1, 145–157 (2004; Zbl 1065.35203)] is also discussed. The results of the last chapter are based on [M. R. Ebert, R. A. Kapp and J. R. Dos Santos Filho, J. Math. Anal. Appl. 359, No. 1, 181–196 (2009; Zbl 1178.35260)]. Bounds for the optimal loss of regularity in the Sobolev scale for the Cauchy problem under consideration are presented here. To conclude, in the last section of chapter four (and the book), the authors present some remarks about this chapter and describe open problems related with well-posed Cauchy problems for weakly hyperbolic operators.

Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)

### MSC:

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

35G05 | Linear higher-order PDEs |

35A10 | Cauchy-Kovalevskaya theorems |

35L45 | Initial value problems for first-order hyperbolic systems |

35L60 | First-order nonlinear hyperbolic equations |