Periodic processes and non-autonomous evolution equations with time- periodic terms
type: Item
instance of: scholarly article
title: Periodic processes and non-autonomous evolution equations with time- periodic terms
zbMATH Open document ID: 0820.47073
DOI: 10.12775/TMNA.1994.020
published in: Topological Methods in Nonlinear Analysis
publication date: 1995-09-06T00:00:00Z
review text: The starting point in the authors' consideration is a Cauchy type problem \[ u_ t= A(u, t), \tag{1} \] where \(A\) is a time dependent mapping of some Banach space \(E\) into itself, subject to some constraints; among others it is \(p\)-periodic i.e. \(A(u, t+ p)= A(u, t)\). It is assumed that the solution to (1) is described by a family \(U(t,\tau)\), \(t\geq \tau\) of (nonlinear) mappings from \(E\) into itself, such that \(u(t)= U(t, \tau) u(\tau)\) solves the initial value problem (1) with \(u(\tau)\) the initial value at \(t= \tau\); under the periodicity condition also(2) \(U(t, \tau)= U(t+ p, \tau+ p)\) holds. In order to study attractors and related concepts of the flow generated by (1) the authors adopt a path which differs from the familiar approach via Poincaré map. They define a semigroup \(S(t)\), \(t\in \mathbb{R}\) on \(E\times T^ 1\), with \(T^ 1\) the circle with length \(p\), according to \[ S(t)(u,\delta)= (U(\delta+ t) u, \delta+ t\text{ mod } p).\tag{2} \] Attractors are then studied in terms of \(S(t)\). In an introductory section, the authors disregard the Cauchy problem (1) and study suitable families \(U(t, \tau)\), \(t\geq \tau\) in their own right. They call a family of mappings \(U(t, \tau)\), \(t\geq \tau\) from \(E\) into \(E\) a ``process'' if, besides some continuity assumptions, the usual causality assumption is satisfied; if moreover (2) holds, the process is called \(p\)-periodic. More generally it is admitted that the process depends on a parameter \(\sigma\) which varies over a metrical space \(\Sigma\), subject to some conditions. In particular a semigroup \(T(t)\), \(t\geq 0\) is given which acts on \(\Sigma\). It is assumed that the following relation holds: \[ U_ \sigma(t+ s, \tau+ s)= U_{T(s)\sigma}(t, \tau).\tag{3} \] Under this assumption a semigroup \(S(t)\), operating on \(E\times \Sigma\), is defined according to \[ S(t)(u, \sigma)= (U_ \sigma(t, 0) u, T(t)\sigma).\tag{4} \] Under certain continuity and compactness assu…
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 58D07
zbMATH DE Number: 733356
zbMATH Keywords: fractal
zbMATH Keywords: Cauchy type problem
zbMATH Keywords: attractors
zbMATH Keywords: semigroup
zbMATH Keywords: compactness
zbMATH Keywords: Hausdorff dimension
author: Vladimir V. Chepyzhov
author: Mark Iosifovich Višik
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
full work available at URL: tmna.1994.020
OpenAlex ID: W2589006662
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description: scientific article; zbMATH DE number 733356