An asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearities
type: Item
instance of: scholarly article
title: An asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearities
zbMATH Open document ID: 1068.35180
published in: Journal of Functional Analysis
publication date: 2005-01-26T00:00:00Z
review text: An asymptotic of the following ordinary differential operator (ODO) with operator coefficients and with nonlinear operators \(N_0,N_1,\dots,N_n\) \[ A(D_t)u(t)=\sum^n_{j=0}{\mathcal D}^{n-j}_tN_j(t;u(t),\dots,D^{\ell-n}_tu(t)),\quad t\in\mathbb R,\tag{1} \] \(\ell>n\), is developed. The conditions on \(A\) and \(N_j\) include elliptic and parabolic PDO. The operators \(N_j\) are defined in appropriate Sobolev spaces \(W^{\ell-n}_{p,\text{loc}}(\mathbb{R};\{B_k\}^{\ell-n}_{k=0})\) and \(B_k\) are Banach spaces. It is shown that for a certain class of equations (1) the problem of the asymptotic behaviour of \(u\) as \(t\to \infty\) can be reduced to that for a finite dimensional dynamical system perturbed by a nonlocal operator, namely \[ \frac d{dt}h(t)+\mathbb{N}(t;h(t))+\mathbb K[h](t)=0,\tag{2} \] where \(\mathbb K\) is a nonlocal nonlinear operator and the components \(\mathbb N_k\) of \(\mathbb N\) are written explicitly. Under some natural assumptions on the nonlinearities \(N_j\) a beautiful representation formula for \((u,\dots,D^{\ell-n-1}_tu)\) is given. It can be viewed as a very interesting generalization of the variation of constants formula in the classical theory of ODO.
Mathematics Subject Classification ID: 35R20
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 2131178
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: Banach spaces
zbMATH Keywords: nonlocal nonlinear operator
zbMATH Keywords: representation formula
zbMATH Keywords: asymptotic theory
zbMATH Keywords: perturbed dynamical system
author: V. A. Kozlov
author: V. Maz'ya
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
full work available at URL: j.jfa.2004.04.004
OpenAlex ID: W2146381483
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DOI: 10.1016/J.JFA.2004.04.004
description: scientific article; zbMATH DE number 2131178