Some applications of the Łojasiewicz gradient inequality

Pages: 2417 - 2427, Volume 11, Issue 6, November 2012

doi:10.3934/cpaa.2012.11.2417       Abstract        References        Full Text (120.7K)       Related Articles

Alain Haraux - UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France (email)

Abstract: In the present survey paper, basic convergence results for gradient-like systems relying on the Łojasiewicz gradient inequality are recalled in a self-contained way. A uniform version of the gradient inequality is used to get directly convergence and the rate of convergence in one step and a new technical trick, consisting in the evaluation of the integral of the velocity norm from $t$ to $2t$ is introduced. A short idea of the state of the art without technical details is also given.

Keywords:   Łojasiewicz inequality, convergence, equilibrium solutions, rate of decay
Mathematics Subject Classification:  34D05, 34D20, 34G20, 37B25, 37L15

Received: October 2010;      Revised: November 2010;      Published: April 2012.

 References