Rate of convergence to equilibrium and Łojasiewicz-type estimates (Q1692106)

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 \[ \ddot u+G(u,\dot u)+\nabla E(u)=0. \]