A compact dissipative dynamical system for a difference equation with diffusion
type: Item
instance of: scholarly article
title: A compact dissipative dynamical system for a difference equation with diffusion
zbMATH Open document ID: 0890.39005
author: L. Fichmann
published in: Journal of Differential Equations
publication date: 1998-04-15T00:00:00Z
review text: A discrete-time version of the parabolic scalar equation is investigated. Consider a sequence \(\{u_n\}_{n\in\mathbb{N}}\) of functions \(u_n\in H^1_0=\{p:[0,1]\to\mathbb{R}: p\) is absolutely continuous with \(p(0)=p(1)=0\) and \(p'\in L_2(0,1)\}\) satisfying: \[ u_{n+1}-u_n=\varepsilon\Delta u_{n+1}+\varepsilon f\circ u_n,\quad n\in\mathbb{N},\;\Delta=\partial^2/\partial x^2,\text{ for given }\varepsilon>0. \tag{1} \] The aim of the paper is to study the iteration properties of the map \(\Phi_\varepsilon:H^1_0\to H^1_0\), \(\Phi_\varepsilon=(I-\varepsilon\Delta)^{-1}\circ(I+\varepsilon\hat f)\), where \(\hat f(p)=f\circ p\) for \(p\in H^1_0\) and \(I\) is the identity of \(H^1_0\) . Remark that the sequence \(\{u_n\}_{n\in\mathbb{N}}\) satisfies (1) if and only if \(u_{n+1}=\Phi_\varepsilon(u_n)\), \(n\in\mathbb{N}\).
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 1054113
zbMATH Keywords: discrete time parabolic equations
zbMATH Keywords: dissipative dynamical system
zbMATH Keywords: difference equation with diffusion
reviewed by: Alexander Mednykh
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3259
OpenAlex ID: W2054375686
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DOI: 10.1006/JDEQ.1997.3259
description: scientific article; zbMATH DE number 1054113