Some applications of the Łojasiewicz gradient inequality
The survey paper presents convergence results for gradient-like systems \[ u'+\nabla{F}(u)=0, \] where \(F\in C^2(\mathbb{R}^N,\mathbb{R})\). If \(u(t)\) is a bounded solution to this equation for \(t\geq 0\), then \[ \lim_{t\to+\infty}\text{ dist}(u(t),\mathcal{E})=0,\text{ where }\mathcal{E}=\{a\in\mathbb{R}^N|\nabla{F}=0\}. \] \textit{S. Łojasiewicz} proved a convergence result for analytic \(F\) [Ensembles semi-analytiques. Bures-sur-Yvette: Institut des Hautes Etudes Scientifiques (1965)]. Using a uniform version of the gradient inequality and deriving first energy estimates the author obtains directly convergence and the rate of convergence. A short idea of the state of the art without technical details is also given.
scientific article; zbMATH DE number 6171012