Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity
type: Item
instance of: scholarly article
title: Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity
zbMATH Open document ID: 1167.34017
author: L. Chergui
published in: Journal of Dynamics and Differential Equations
publication date: 2008-08-20T00:00:00Z
review text: Given the initial value problem \[ U^{\prime\prime}(t)+\| U^{\prime}(t)\|^{\alpha}U^{\prime}(t)+\nabla F(U(t))=0,\quad U(0)=U_0\in\mathbb{R}^N, \quad U^{\prime}(0)=U_1\in\mathbb{R}^N, \] where \(N\geq1\), \(\alpha>0\) and \(F:\mathbb{R}^N\rightarrow\mathbb{R}\) is a function of class \(C^2\). Introduce \(S=\{x\in\mathbb{R}^N:\nabla F(x)=0\}\) and assume that there exists \(\theta\in(0,1/2)\) such that for every \(a\in S\) there exists \(\sigma_a>0\) such that \(\|\nabla F(t)\|\geq| F(u)-F(a)|^{1-\theta}\) for all \(u\)'s with \(\| u-a\|\leq\sigma_a\). Fix \(\alpha\in[0,\theta/(1-\theta))\) and assume that \(U\in W^{2,\infty}\) is a solution of the initial value problem. Then, there exists \(C>0\) such that for all \(t>0\) one has \(\| U^{\prime}(t)\|+\| U(t)-a\|\leq C\cdot(1+t)^{-\eta}\), where \(\eta=\frac{\theta-(1-\theta)\alpha}{1-2\theta+(1-\theta)\alpha}\) (Theorem 1.3). In particular, \[ \lim_{t\rightarrow+\infty}\| U^{\prime}(t)\|+\| U(t)-a\|=0\quad\text{(Theorem 1.2)}. \] The proofs are based on several differential inequalities of first order and on the Lojasiewicz gradient inequality.
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 5314308
zbMATH Keywords: gradient system
zbMATH Keywords: equilibrium
zbMATH Keywords: rate of convergence
reviewed by: Octavian Mustafa
MaRDI profile type: Publication
full work available at URL: s10884-007-9099-5
OpenAlex ID: W2006286580
cites work: Q3997729
cites work: Convergence of solutions to second-order gradient-like systems with analytic nonlinearities
cites work: Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity
cites work: Q3952851
DOI: 10.1007/S10884-007-9099-5
description: scientific article; zbMATH DE number 5314308