Integral manifolds for semilinear evolution equations and admissibility of function spaces
type: Item
instance of: scholarly article
title: Integral manifolds for semilinear evolution equations and admissibility of function spaces
zbMATH Open document ID: 1272.34081
author: Nguyen Thieu Huy
author: Vu Thi Ngoc Ha
author: Phi Ha
published in: Ukrainian Mathematical Journal
publication date: 2013-01-23T00:00:00Z
full work available at URL: http://dspace.nbuv.gov.ua/handle/123456789/164414
review text: The authors prove the existence of stable, unstable and center integral manifolds to the integral equation \[ u(t)=U(t,s)u(s)+\int_s ^t U(t,\xi)f(\xi,u(\xi)\,\mathrm{d}\xi\qquad \text{for a.e. }t\geq s,\;t,s\in \mathbb R_+, \] where the evolution familiy \((U(t,s)_{t\geq s\geq 0}\) possesses an exponential trichotomy on \(\mathbb R_+\). Instead of a classical Lipschitz condition of the type \[ \| f(t,x)-f(t,y)\|\leq q\,\| x-y\| \] and sufficiently small \(q\), they assume a \(\phi\)-Lipschitz property \[ \| f(t,x)-f(t,y)\|\leq \phi (t)\,\| x-y\| \] with \(\phi\) belonging to suitable classes of admissible functions, where the smallness of \(q\) is replaced by the smallness of \(\sup_{t\geq 0}\int _t ^{t+1}\phi (\tau)\,\mathrm{d}\tau\).
reviewed by: Norbert Koksch
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34C45
zbMATH DE Number: 6128276
zbMATH Keywords: integral manifold
zbMATH Keywords: exponential trichotomy
zbMATH Keywords: Lyapunov-Perron method
zbMATH Keywords: admissibility of function spaces
MaRDI profile type: Publication
OpenAlex ID: W2021366779
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DOI: 10.1007/S11253-012-0686-1
description: scientific article; zbMATH DE number 6128276