Rate of convergence to equilibrium and Łojasiewicz-type estimates
type: Item
instance of: scholarly article
title: Rate of convergence to equilibrium and Łojasiewicz-type estimates
zbMATH Open document ID: 1394.34099
author: Tomáš Bárta
published in: Journal of Dynamics and Differential Equations
publication date: 2018-01-26T00:00:00Z
review text: The paper deals with the differential equation \[ \dot u+F(u)=0, \,F:M\to TM, \] where \((M,g)\) is a smooth Riemannian manifold with metric \(g\), and \(TM\) is the tangent bundle. The author formulates sufficient conditions for the convergence of the solutions of a gradient-like system to a point in the omega-limit set \(\omega(u)=\{\varphi\in M:\exists t_n \nearrow +\infty, u(t_n)\to\varphi\}\). The main results of the paper provide estimates of the convergence rate under a generalized Łojasiewicz condition, and, as a corollary, under an angle condition and the Kurdyka-Łojasiewicz inequality. Furthermore, the obtained abstract result is applied to the second-order equation with damping of the form \[ \ddot u+G(u,\dot u)+\nabla E(u)=0. \]
reviewed by: V. Grushkovskaya
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 6829893
zbMATH Keywords: gradient-like system
zbMATH Keywords: rate of convergence
zbMATH Keywords: Kurdyka-Łojasiewicz inequality
zbMATH Keywords: Łojasiewicz condition
zbMATH Keywords: damped second order equation
MaRDI profile type: Publication
full work available at URL: s10884-016-9549-z
OpenAlex ID: W2511988368
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DOI: 10.1007/S10884-016-9549-Z
description: scientific article; zbMATH DE number 6829893