Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model

Zhuchun Li; Yi Liu; Xiaoping Xue

doi:10.3934/dcds.2019014

Article Contents

2019, Volume 39, Issue 1: 345-367. Doi: 10.3934/dcds.2019014

This issue Previous Article The variational discretization of the constrained higher-order Lagrange-Poincaré equations Next Article Qualitative properties of positive solutions for mixed integro-differential equations

Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model

1.
Department of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China
2.
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China
3.
Department of Mathematics, Suihua University, Suihua 152000, China

* Corresponding author: Xiaoping Xue
* Corresponding author: Xiaoping Xue

Received: December 31, 2017

Revised: June 30, 2018

Published: December 2018

This work was supported by NSF of China grants 11731010 and 11671109

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The existence and uniqueness/multiplicity of phase locked solution for continuum Kuramoto model was studied in [12,29]. However, its asymptotic behavior is still unknown. In this paper we concern the asymptotic property of classic solutions to continuum Kuramoto model. In particular, we prove the convergence towards a phase locked state and its stability, provided suitable initial data and coupling strength. The main strategy is the quasi-gradient flow approach based on Łojasiewicz inequality. For this aim, we establish a Łojasiewicz type inequality in infinite dimensions for continuum Kuramoto model which is a nonlocal integro-differential equation. General theorems for convergence and stability of (generalized) quasi-gradient system in an abstract setting are also provided based on Łojasiewicz inequality.

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Mathematics Subject Classification: Primary: 92D25, 34D06, 46N20; Secondary: 34K25.

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