The Mathieu-Hill operator equation with dissipation and estimates of its instability index
type: Item
instance of: scholarly article
title: The Mathieu-Hill operator equation with dissipation and estimates of its instability index
zbMATH Open document ID: 0913.35019
DOI: 10.1007/BF02354989
author: S. V. Zelik
published in: Mathematical Notes
publication date: 1999-05-17T00:00:00Z
review text: The behaviour as \(t\to\infty\) of solutions of the equation \[ {\partial_t}^2+\gamma\partial_t u+Au+f(t)u=0 \] in a Hilbert space is studied, where \(A=A^*\) is a positive definite operator with compact inverse, and the operator \(f\) is periodic in \(t\). The notion of instability index is introduced for this equation. It is proved that the instability index is finite under natural assumptions (\(f\) must be dominated by \(A\)). Asymptotic estimates of the instability index are obtained as \(\gamma\to 0\), and an example is constructed showing that they cannot be improved. Furthermore, the qualitative characteristics of the spectrum of the monodromy operator and the existence of the Floquet representation for this problem is studied.
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 1237995
zbMATH Keywords: large time asymptotics
zbMATH Keywords: Floquet representation
zbMATH Keywords: spectrum of the monodromy operator
reviewed by: Ivan Ginchev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1237995