Asymptotic behavior of solutions to Eells-Sampson equations near stable harmonic maps
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions to Eells-Sampson equations near stable harmonic maps
zbMATH Open document ID: 0688.58007
DOI: 10.1007/BF01159967
author: Hisashi Naito
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174098
review text: The i.v. problem (ES) \(\partial f/\partial t=\Delta f\) on \(M\times [0,\infty)\), \(f(0)=f_ 0\) on M is considered. The following theorem is proved: Theorem. Let (M,g) and (N,h) be closed Riemannian manifolds, and F: \(M\to N\) be a stable harmonic map. There exists a positive number \(\epsilon\) such that if \(\| F-f_ 0\|_{H^ m(M,N)}<\epsilon\) for some \(m>\dim M+2\), then (ES) with initial value \(f_ 0\in H^ m(M,N)\) has a unique global solution in \(C^{1/2}([0,\infty);H^{m- 1}(M,N))\cap C^ 0([0,\infty);H^ m(M,N))\), and converge to F in \(H^ m(M,N)\) with exponential order as \(t\to \infty\).
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 4128100
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: stable harmonic map
MaRDI profile type: Publication
cites work: Quasilinear Evolution Equations and Parabolic Systems
cites work: A Report on Harmonic Maps
cites work: Q3662316
cites work: Harmonic Mappings of Riemannian Manifolds
cites work: Weak and strong solutions of the Navier-Stokes initial value problem
cites work: Solutions for semilinear parabolic equations in \(L^ p\) and regularity of weak solutions of the Navier-Stokes system
cites work: On Homotopic Harmonic Maps
cites work: Harmonic maps of manifolds with boundary
cites work: La formule de la variation seconde de l'energie au voisinage d'une application harmonique
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Global existence of small solutions to a class of nonlinear evolution equations
cites work: The linearization principle for the stability of solutions of quasilinear parabolic equations. I
cites work: Asymptotics for a class of non-linear evolution equations, with applications to geometric problems
cites work: The Second Variation Formula for Harmonic Mappings
OpenAlex ID: W2019916617
description: scientific article; zbMATH DE number 4128100