# American Institute of Mathematical Sciences

January  2019, 39(1): 345-367. doi: 10.3934/dcds.2019014

## 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

Received  January 2018 Revised  July 2018 Published  October 2018

Fund Project: This work was supported by NSF of China grants 11731010 and 11671109.

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.

Citation: Zhuchun Li, Yi Liu, Xiaoping Xue. Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 345-367. doi: 10.3934/dcds.2019014
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