Floquet theory for parabolic differential equations
type: Item
instance of: scholarly article
title: Floquet theory for parabolic differential equations
zbMATH Open document ID: 0804.35045
author: Shui-Nee Chow
author: Kening Lu
author: John Mallet-Paret
published in: Journal of Differential Equations
publication date: 1994-06-15T00:00:00Z
review text: The Floquet theorem is extended to the scalar parabolic equation \[ u_ t = u_{xx} + a(t,x) u_ x + b(t,x)u, \quad 0<x< \pi \] with Dirichlet boundary conditions \(u(t,0) = (t, \pi) = 0\) on the interval \((0,\pi)\). The coefficients \(a(t,x)\), \(b(t,x)\) are assumed to be \(T\)-periodic in \(t\) for all \(x\), and sufficiently smooth. Considering this problem as a system in the Hilbert space \(H=L^ 2(0,\pi)\) the existence of a time- dependent linear isomorphism \(v=\Phi(t)u\) is proved which is \(T\)-periodic as well as continuous in \(t\) in the operator norm and which transforms this problem to the autonomous system \[ v_ t = v_{xx} + q(x)v, \quad v(t,0) = v(t,\pi) = 0 \] for some function \(q(x)\).
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35A22
zbMATH DE Number: 589984
zbMATH Keywords: Floquet theory
zbMATH Keywords: periodic invariant manifolds
zbMATH Keywords: scalar parabolic equation
reviewed by: Josef Diblík
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1047
OpenAlex ID: W1980356323
DOI: 10.1006/JDEQ.1994.1047
description: scientific article; zbMATH DE number 589984