A class of locally solvable semilinear equations of weakly hyperbolic type
type: Item
instance of: scholarly article
title: A class of locally solvable semilinear equations of weakly hyperbolic type
zbMATH Open document ID: 0849.35076
DOI: 10.1007/BF01759266
author: Piero D'ancona
author: Renato Manfrin
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1996-07-08T00:00:00Z
review text: The goal of the present paper is to study the following quasilinear weakly hyperbolic Cauchy problem \[ u_{tt}- \sum^n_{i, j= 1} (a_{ij}(t, x) u_{x_i})_{x_j}+ \sum^n_{j= 1} b_j(t, x) u_{x_j}= f(u, u_t), \] \[ u(0, x)= u_0(x),\;u_t(0, x)= u_1(x). \] Here, weakly hyperbolic means that \(a_{ij}= \overline{a_{ji}}\), \(\Lambda^2 |\xi|^2\geq \sum^n_{j= 1} a_{ij} \xi_i \xi_j\geq 0\). In general one cannot expect well-posedness in \(C^\infty\) if the coefficients, data and right-hand side are only \(C^\infty\). For this reason one has to prescribe so-called Levi conditions. The authors use the Oleijnik condition \[ B \Biggl( \sum^n_{j= 1} b_j(t, x) \xi_j\Biggr)^2\leq A \sum^n_{i, j= 1} a_{ij}(t, x) \xi_i \xi_j+ \sum^n_{i, j= 1} \partial_t a_{ij}(t, x) \xi_i \xi_j \] with some positive constants \(A\) and \(B\). Then they can prove local \(C^\infty\)-well posedness under some additional standard assumptions. Here well posedness means local existence and finite speed of propagation, cone of dependence respectively. Two different proofs are presented. If \(f\) is nonlinear only in \(u\), then some a-priori estimates and an iteration process give the existence result. If \(f\) is nonlinear in \(u_t\), too, then we have in the a-priori estimates some uniformly loss of Sobolev regularity. Consequently, the application of the Nash-Moser technique leads to the desired results.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L80
zbMATH DE Number: 878454
zbMATH Keywords: quasilinear weakly hyperbolic Cauchy problem
zbMATH Keywords: well-posedness
zbMATH Keywords: Nash-Moser technique
reviewed by: Michael Reissig
MaRDI profile type: Publication
cites work: An example of a weakly hyperbolic Cauchy problem not well posed in \(C^\infty\)
cites work: Sur les problèmes de Cauchy hyperboliques bien posés
cites work: Racine carrée d'une fonction différentiable
cites work: The inverse function theorem of Nash and Moser
cites work: A necessary and sufficient condition for the hyperbolicity of second order equations with two independent variables
full work available at URL: https://doi.org/10.1007/bf01759266
OpenAlex ID: W1979425184
description: scientific article; zbMATH DE number 878454