Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem
type: Item
instance of: scholarly article
title: Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem
zbMATH Open document ID: 1097.35092
author: Massimo Cicognani
author: Ferruccio Colombini
published in: Journal of Differential Equations
publication date: 2006-03-20T00:00:00Z
review text: Let the operator \(P\) in \([0, T]\times\mathbb{R}^n_x\) be defined by \[ P= \partial^2_t+ 2iB(t,x,D_x)\partial_t+ A(t,x,D_x), \] \(D_x= -i\nabla_x\), with real symbols \[ A(t,x,\xi)= \sum^n_{j,k=1} a_{jk}(t, x)\xi_j \xi_k,\;B(t,x,\xi)= \sum^n_{j=1} b_j(t, x)\xi_j \] that satisfy the condition of strict hyperbolicity \[ B^2(t,x,\xi)+ A(t,x,\xi)\geq c_0|\xi|^2\quad (c_0> 0) \] for \(t\in [0, T]\), \(x,\xi\in\mathbb{R}^n\). Given a continuous function \(\omega(\theta)\) such that \(\omega(\theta)\downarrow 0\), for \(\theta\to 0^+\). Denote by \(M^\omega([0, T])\) the space of all functions \(a(t)\) such that \[ |a(t+\tau)- a(t)|\leq C|\tau||\log|\tau|)\quad (C> 0) \] for \(t,t+ \tau\in [0, T]\), \(0\leq|\tau|\leq{1\over 2}\). Let the coefficients of \(P\) be \(a_{jk}\), \(b_j\in M^\omega([0,T]:{\mathfrak B}^\infty)\). Then it proved that for every \(s,\delta> 0\) there is \(C_{s,\delta}> 0\) such that \[ \| u(t)\|_{H^{s+1-\delta}}+ \|\partial_t u(t)\|_{H^{s-\delta}}\leq C\Biggl(\| u(0)\|_{H^{s+1}}+ \|\partial_t u(0)\|_{H^s}+ \int^t_0\| Pu(\tau)\,d\tau\Biggr) \] for \(t\in[0, T]\), and for all \(u\in\bigcap^2_{j=0} C^j([0, T], H^{s+2-j})\).
reviewed by: Kunihiko Kajitani
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 5013575
zbMATH Keywords: Levi condition
zbMATH Keywords: log-Lipschitz continuity
MaRDI profile type: Publication
full work available at URL: j.jde.2005.06.019
OpenAlex ID: W1994780266
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DOI: 10.1016/J.JDE.2005.06.019
description: scientific article; zbMATH DE number 5013575