Blowup for nonlinear hyperbolic equations
type: Item
instance of: scholarly article
title: Blowup for nonlinear hyperbolic equations
zbMATH Open document ID: 0820.35001
author: S. Alinhac
published in: Progress in Nonlinear Differential Equations and Their Applications
publication date: 1995-05-02T00:00:00Z
review text: These lecture notes give a short and clear introduction to the problem of singularity formation in nonlinear hyperbolic problems, by one of the active researchers in this rapidly growing field. Coverage includes typical blow-up criteria, estimates of the maximal time of existence by asymptotic methods, but first and foremost general strategies for establishing the blow-up mechanism. Indeed, recent work has shown that singularity formation can be understood by performing appropriate changes of variables acting on dependent and independent variables; the singularities of the solution are simply due to the properties of these changes of variables. These techniques form the unifying theme of the book. They enable one to go, for the first time with this level of generality, beyond estimates of the time of the first singularity. This new methodology seems to have permanent value. The style is both lively and rigorous, and the proofs are well-organized making the main ingredients transparent. The examples are concrete and well-chosen. The material is at the forefront of current research, and the open problems are clearly introduced. One may mention that the questions mentioned in the book relative to the reviewer's work have been answered since the book was written: the results do extend to the non-analytic case, and they do describe the correct blow-up mechanism for data close to constants. As the author mentions, his work does not aim to be encyclopaedic; it nevertheless is a very good snapshot of the subject. This book is very stimulating and enjoyable. It should encourage students and researchers alike to contribute to the subject.
reviewed by: Satyanad Kichenassamy
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35B20
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35A20
zbMATH DE Number: 750177
zbMATH Keywords: singularity formation
zbMATH Keywords: nonlinear hyperbolic problems
zbMATH Keywords: blow-up criteria
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 750177