Lifespan and blow-up of solutions of the quasilinear wave equations in two dimensions. II
type: Item
instance of: scholarly article
title: Lifespan and blow-up of solutions of the quasilinear wave equations in two dimensions. II
zbMATH Open document ID: 0844.35102
DOI: 10.1215/S0012-7094-94-07322-5
author: S. Alinhac
published in: Duke Mathematical Journal
publication date: 1996-09-03T00:00:00Z
review text: [For part I see Ann. Sci. Ec. Norm. Super., IV. Ser. 28, No. 2, 225-251 (1995).] The author considers the quasilinear wave equation in two dimensions, and demonstrates the existence of a function \(T^a_\varepsilon\) named ``asymptotic life time'' of the solution, with the two properties i) \(\forall N\in \mathbb{N}\), \(T_\varepsilon\geq T^a_\varepsilon- \varepsilon^N\), for \(0< \varepsilon\leq \varepsilon_N\). ii) There is \(C> 0\) so that \(1/C(T_\varepsilon- t)\leq |\nabla^2 u|_{L^\infty}\leq C(1/(T^a_\varepsilon- t))\), where \(u\) is the solution of the wave equation, \(T_\varepsilon\) is the life time and \(\varepsilon> 0\) the size parameter. The result is obtained in two steps: First, an approximate solution is constructed. Then it is proved that the real solution has almost the same life time as the approximate solution and an estimation of the difference between real and approximate solutions is given by using the energy inequality.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 572043
zbMATH Keywords: wave equation
zbMATH Keywords: life time
zbMATH Keywords: energy inequality
reviewed by: Valeriu Al. Sava
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 572043