Global smooth solvability of a nonlinear boundary value problem for a quasilinear hyperbolic system
type: Item
instance of: scholarly article
title: Global smooth solvability of a nonlinear boundary value problem for a quasilinear hyperbolic system
zbMATH Open document ID: 0573.35057
DOI: 10.1007/BF01066548
author: S. Ya. Aliev
author: Yu. L. Majstrenko
published in: Ukrainian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The paper deals with the system (1) \(u_ t+uu_ x=0\), \(v_ t-vv_ x=0\) for \(0\leq x\leq 1\), \(t\geq 0\), with boundary conditions \((2)\quad u(0,t)=v(0,t),\quad v(1,t)=\phi (u(1,t)),\) \(t\geq 0\), and usual initial conditions. Using the method of characteristics the problem is transformed to the functional equation \(z(t+z(t))=\psi (z(t)),\quad \psi (z)=2/\phi (2/z),\) which is used for the proof of global existence and uniqueness of the continuously differentiable solution of the problem for a certain class of functions \(\phi\) and initial conditions. The asymptotic behaviour of the solution is studied. Analogous results are proved for the system (1) with perturbed boundary condition \((2')\quad u(0,t)=v(0,t),\quad \epsilon v_ t(1,t)+v(1,t)=\phi (u(1,t)),\) \(t\geq 0\), \(\epsilon >0\).
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35L50
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 3914840
zbMATH Keywords: quasilinear hyperbolic system
zbMATH Keywords: method of characteristics
zbMATH Keywords: functional equation
zbMATH Keywords: global existence
zbMATH Keywords: uniqueness
zbMATH Keywords: continuously differentiable solution
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: perturbed boundary condition
reviewed by: Alexandr Doktor
MaRDI profile type: Publication
cites work: Q3909393
cites work: Q5592596
cites work: Q4151274
description: scientific article; zbMATH DE number 3914840