Some well-posed Cauchy problem for second order hyperbolic equations with two independent variables
type: Item
instance of: scholarly article
title: Some well-posed Cauchy problem for second order hyperbolic equations with two independent variables
zbMATH Open document ID: 1244.35082
author: Ferruccio Colombini
author: Tatsuo Nishitani
author: Nicola Orrù
author: Ludovico Pernazza
published in: Osaka Journal of Mathematics
publication date: 2011-11-08T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1317044940
review text: The authors consider the \(C^\infty\) well-posedness for the second-order hyperbolic equation \(Pu=\partial^2_t u- a(t,x)\partial^2_x u= f\) with two independent variables \((t, x)\). They prove that if the \(C^\infty\) function \(a(t, x)\geq 0\) satisfies \(\partial^p_t a(0,0)\neq 0\) with some \(p\) and if the discriminant of \(a(t, x)\) vanishes of finite order at \(x=0\), the Cauchy problem for \(P\) is \(C^\infty\) well-posed in a neibourhood of the origin. As a generalization of the above result they prove that if \(a(t, x)= \prod^r_{j=1} b_j(t,x)^{m_j}\) with positive integers \(m_j\), where each \(b_j(t, x)\) satisfies \(\partial^{p_j}_t b_j(0,0)\neq 0\), \(b_j(t,x)^{m_j}\geq 0\) and the discriminant of \(b_j\) vanishes of finite order, then the Cauchy problem for \(P\) is \(C^\infty\) well-posed in a neigborhood of the origin.
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35L80
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 5969043
zbMATH Keywords: Cauchy problem
zbMATH Keywords: \(C^\infty\) well posedness
zbMATH Keywords: hyperbolic equation
reviewed by: Kunihiko Kajitani
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5969043