# American Institute of Mathematical Sciences

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November  2012, 11(6): 2417-2427. doi: 10.3934/cpaa.2012.11.2417

## Some applications of the Łojasiewicz gradient inequality

 1 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris

Received  October 2010 Revised  November 2010 Published  April 2012

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.
Citation: Alain Haraux. Some applications of the Łojasiewicz gradient inequality. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2417-2427. doi: 10.3934/cpaa.2012.11.2417
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##### References:
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