Existence and non-existence of global smooth solutions for \(p\)-system with relaxation
type: Item
instance of: scholarly article
title: Existence and non-existence of global smooth solutions for \(p\)-system with relaxation
zbMATH Open document ID: 0958.35081
author: Tong Yang
author: Changjiang Zhu
published in: Journal of Differential Equations
publication date: 2000-08-28T00:00:00Z
review text: The authors consider the Cauchy problem \(v_t-u_x=0\), \(u_t+p(v)_x ={1\over \varepsilon}(f(v)-u)\), with the initial conditions: \((v,u)(x,0)= (v_0(x), u_0(x))\), where \(\varepsilon>0\) is the relaxation time, \(p(v)\) satisfies several stated conditions. The initial conditions yield that the system has the eigenvalues \(\lambda(v)= -\sqrt{-p'(v)}\), \(\mu(v)= \sqrt{-q'(v)}\), and with the substitutions \(r=u+\Phi(v)\), \(s=u-\Phi(v)\), where \(\Phi(v)= \int_v^{v_0(x)} \mu(\tau) d\tau\), the above system is transformed to this diagonal form \(r_t+\lambda (v)r_x= -{1\over 2\varepsilon} (r+s-F(r-s))\), \(s_t+\mu(v) s_x=-{1\over 2\varepsilon} (r+s-F(r-s))\), with the transformed initial conditions. With this change the authors prove the existence of global smooth solutions to the original system.
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35L45
zbMATH DE Number: 1437330
zbMATH Keywords: transformation into diagonal form
reviewed by: Hussain S. Nur
MaRDI profile type: Publication
full work available at URL: jdeq.2000.3710
OpenAlex ID: W1970106512
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DOI: 10.1006/JDEQ.2000.3710
description: scientific article; zbMATH DE number 1437330