Convergence and stability of generalized gradient systems by łojasiewicz inequality with application in continuum kuramoto model

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 ow approach based on Lojasiewicz inequality. For this aim, we establish a Lojasiewicz 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 Lojasiewicz inequality.