Blow-up mechanism of classical solutions to quasilinear hyperbolic systems
type: Item
instance of: scholarly article
title: Blow-up mechanism of classical solutions to quasilinear hyperbolic systems
zbMATH Open document ID: 1132.35415
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-06-06T00:00:00Z
review text: The Cauchy problem for the first order hyperbolic system \[ \frac{\partial u} {\partial t} + A(u)\frac{\partial u} {\partial x} = 0, \qquad u(x,0)= \varphi(x), \tag{1} \] where \( u =(u_1 ,u_2,\dots,u_n)^T \) is unknown vector function of \((t,x)\), \(A(u)\) is an \( n \times n \) matrix with suitably smooth elements \( a(u) \) \((i,j=1,\dots,n)\) and \( \varphi (x)=(\varphi_1(x),\dots,\varphi _n(x))^T \) is a \( C^1 \) vector function of \( x \) is discussed. It is supposed that in a neighborhood of \( u = 0 \) system is strictly hyperbolic: \( A(u)\) has \(n\) distinct real eigenvalues \( \lambda _1(u)< \lambda _2(u)<\cdots<\lambda_n(u)\). The blowup phenomenon of the classical solutions to the problem (1), particularly, the geometric blowup mechanism in the noncritical case for general initial data satisfying \[ \sup_{x \in\mathbb R}\{(1+| x|^{1+\mu}(|\varphi(x)|+ |\varphi'(x)|))\}\ll1 \] is studied. The estimate of lifespan and under the additional hypothesis sharp estimate of the \( C^1 \) solution is obtained. It is proved that the envelope of the same family of characteristics which yields the blowup of classical solutions to the problem (1).
reviewed by: Mersaid Aripov
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 5161908
zbMATH Keywords: estimate of lifespan
author: Li-bin Wang
MaRDI profile type: Publication
full work available at URL: j.na.2006.05.019
OpenAlex ID: W2057828795
cites work: Q3769304
cites work: Q4824228
cites work: Formation of singularities in one-dimensional nonlinear wave propagation
cites work: Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data
cites work: A note on blow-up phenomenon of classical solutions to quasilinear hyperbolic systems
cites work: Q4852353
cites work: Une remarque sur les coordonnées normalisées et ses applications aux systèmes hyperboliques quasi linéaires
cites work: Q3159189
cites work: Breakdown of classical solutions to quasilinear hyperbolic systems
cites work: Global existence of weakly discontinuous solutions to the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems
cites work: Blow-up mechanism of classical solutions to quasilinear hyperbolic systems in the critical case
cites work: Q3762928
cites work: Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems
cites work: Global classical solutions for general quasilinear hyperbolic systems with decay initial data
cites work: Formation of singularities for a kind of quasilinear non-strictly hyperbolic system.
cites work: Pointwise estimates of solutions to Cauchy problem for quasilinear hyperbolic systems
cites work: Global classical solutions with small initial total variation for quasilinear hyperbolic systems
cites work: Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy
DOI: 10.1016/J.NA.2006.05.019
description: scientific article; zbMATH DE number 5161908