Hyperbolic equations with cofficients rapidly oscillating in time: a result of nonstability
type: Item
instance of: scholarly article
title: Hyperbolic equations with cofficients rapidly oscillating in time: a result of nonstability
zbMATH Open document ID: 0576.35009
DOI: 10.1016/0022-0396(84)90131-1
author: Ferruccio Colombini
author: Sergio Spagnolo
published in: Journal of Differential Equations
publication date: 1984-01-01T00:00:00Z
review text: The principle result of this article is a proof of the nonstability of solutions of the hyperbolic equation \[ (1)\quad u_{tt}- \sum^{N}_{i,j=1}\alpha_{ij}(k,t)u_{x_ ix_ j}=0,\quad u(x,0)=\phi (x),\quad u_ t(x,0)=\psi (x) \] as k approaches \(+\infty\). The initial data \(\phi\) and \(\psi\) are assumed analytic, since even in the \(C^{\infty}\) case examples can be given of problems having no solution in the space of distributions. It is proved that for all nonconstant \(L^ 1_{loc}\), periodic coefficients \(\alpha_{ij}(t)\), there exist analytic initial data \(\phi\) and \(\psi\) for which the sequence of solutions \(\{u_ j(\cdot,t)\}\) of (1) becomes unbounded in the space of distributions on \(R^ N.\) The proof uses a technique due to P. Ungar, which involves the study of the Hill's discriminant of the ordinary differential equation \(w''+\beta (t)w=0\), as a holomorphic function. An example is given to illustrate the main result, and it is conjectured that an analogous result holds more generally for a sequence of problems \[ u_{tt}-\Sigma a_{ij,k}(t)u_{x_ ix_ j}=0,\quad u(x,0)=\phi (x),\quad u_ t(x,0)=\psi (x). \]
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 3921042
zbMATH Keywords: rapidly oscillating coefficients
zbMATH Keywords: Cauchy problem
zbMATH Keywords: nonstability
reviewed by: Allan L. Edelson
MaRDI profile type: Publication
full work available at URL: 0022-0396(84)90131-1
OpenAlex ID: W2070925052
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description: scientific article; zbMATH DE number 3921042