Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for a nonhomogeneous quasilinear hyperbolic system
type: Item
instance of: scholarly article
title: Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for a nonhomogeneous quasilinear hyperbolic system
zbMATH Open document ID: 1109.35067
author: De-Xing Kong
author: Tatsien Li
author: Zhi-Qiang Shao
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: The authors of this interesting paper consider a nonhomogeneous quasilinear hyperbolic system \[ {\partial u\over \partial t}+A(u){\partial u\over \partial x}+Lu=0, \] where \(u=(u_1,u_2, \ldots , u_n)^T\) is the unknown vector function of \((t,x)\), \(A(u)\) is \(n\times n\) matrix with suitably smooth elements \(a_{ij}(u)\) (\(i,j=1,2,\ldots , n\)), \(L>0\) is a constant. The following decay initial data at \(t=0\) is given: \[ u=\varphi (x) \quad (x\geq 0), \] where \(\varphi (x)\) is a small \(C^1\) function of \(x\) with certain decay properties as \(x\to\infty \). The main result here is that there exists \(\theta_0>0\) so small that for any given \(\theta \in [0,\theta_0]\), the above stated Cauchy problem admits a unique global \(C^1\) solution \(u=u(t,x)\) on the domain \(D=\{(t,x): t\geq 0\), \(x\geq x_n(t)\}\), where \(x=x_n(t)\) is the \(n\)-th forward characteristic emanating from the origin \(O(0,0)\), \[ {d x_n(t)\over dt}=\lambda_n(u(t,x_n(t))), \qquad x_n(0)=0. \] (\(\lambda_n(u)\) is the \(n\)-th eigenvalue to the system.) Using the above stated theory the authors prove the existence of global classical, \(C^1\) solutions of the flow equations of a model class of fluids with viscosity induced by fading memory with small smooth initial data.
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L45
zbMATH DE Number: 5078920
zbMATH Keywords: weak linear degeneracy
zbMATH Keywords: \(n\)-th forward characteristic
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.01.065
OpenAlex ID: W1984560681
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DOI: 10.1016/J.JMAA.2006.01.065
description: scientific article; zbMATH DE number 5078920