Asymptotic expansions on the small parameter for solutions of linear symmetric systems
type: Item
instance of: scholarly article
zbMATH Open document ID: 0999.35017
author: O. Hawamdeh
publication date: 2002-11-30T00:00:00Z
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 35B20
zbMATH DE Number: 1663581
zbMATH Keywords: symmetric hyperbolic systems
zbMATH Keywords: boundary layer functions
MaRDI profile type: Publication
title: Asymptotic expansions on the small parameter for solutions of linear symmetric systems
published in: Anale Ştiinţifice Facultatea de Matematică şi Informatică. Universitatea de Stat din Moldova
review text: Consider the following Cauchy problem NEWLINE\[NEWLINE (P_0+\varepsilon P_1)U =F(x,t),\quad x\in \mathbb{R}^d,\;t>0,\qquad U(\varepsilon,x,0)=U(x),\quad x\in \mathbb{R}, \tag{1} NEWLINE\]NEWLINE where \(P_i =(A_i\partial _t+B_i(x)\partial _x)+G_i(x)\), \(i=0,1\), \(B_i\), \(G_i\) are real \(n\times n\) matrices, \(d\geq 1\), \(\varepsilon >0\) is a small parameter, \(U,F:\mathbb{R}\times [0,\infty)\to \mathbb{R}^n\), NEWLINE\[NEWLINE A_0=\left(\begin{matrix} I_m & 0\\ 0 & 0 \end{matrix} \right),\quad A_1=\left(\begin{matrix} 0 & 0\\ 0 & I_{n-m} \end{matrix} \right),\quad \quad 0\leq m\leq n, NEWLINE\]NEWLINE and \(I_k\) is a unity matrix. The author studies the behavior of solution \(U(\varepsilon,x,t)\) to the problem \((1)\) as \(\varepsilon\to 0\). For \(s\in N\) denote by \(H^s\) the usual Sobolev spaces with the norm \(\|\cdot\|_s\), \(H^s_n =(H^s)^n\) is the Hilbert space equipped with the scalar product \((f_1,f_2)_{s,n}=\sum _{j=1}^n (f_{1j},f_{2j})_s\), \(f_i =(f_{i1},\dots ,f_{in})\), \(i=1,2\) and with the norm \(\|\cdot \|_{s,n}\) generated by this scalar product. For \(k\in N^*\) and \(1\leq p \leq \infty\) and Banach space \(X\) set \(W^{k,p}(a,b;X)=\{u\in{\mathcal D}'((a,b);X)\); \(u^{(j)}\in L^p(a,b;X)\), \(j=0,1,\dots, k\}\), where \(u^{(j)}\) is the distributional derivative of order \(j\). Denote by \(B=B_0+\varepsilon B_1\), \(G=G_0+\varepsilon G_1\). The special forms of matrices \(A_0\) and \(A_1\) involve the natural representations of matrices \(B_i,G_i\) by blocks NEWLINE\[NEWLINE B_j = \left(\begin{matrix} B_{j1} & B_{j2}\\ B^*_{j2} & B_{j3} \end{matrix} \right),\qquad G_j = \left(\begin{matrix} G_{j1} & G_{j2}\\ G^*_{j2} & G_{j3} \end{matrix} \right),\qquad j=0,1, NEWLINE\]NEWLINE where \(B_{j1}, G_{j1} \in M^m(\mathbb{R})\), \(B_{j2},G_{j2} \in M^{m\times (n-m)}(\mathbb{R})\), \(B_{j3},G_{j3} \in M^{n-m}(\mathbb{R})\), and ``*'' means the transposition. Suppose that \(B\) and \(G\) satisfy the following conditions: \(B_i(x), G_i(x)\), \(i…
reviewed by: Andrei Perjan
description: scientific article; zbMATH DE number 1663581