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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
##### References:
 [1] P.-A. Absil, R. Mahony and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions,, SIAM J. Optim., 16 (2005), 531. Google Scholar [2] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations,, Asymptotic Analysis, 69 (2010), 31. Google Scholar [3] I. Ben Hassen and L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation,, J. Dynam. Diff. Equa., 23 (2011), 315. Google Scholar [4] I. Ben Hassen and A. Haraux, Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy,, J. Funct. Anal., 260 (2011), 2933. Google Scholar [5] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity,, J. Dynam. Diff. Equa., 20 (2008), 643. Google Scholar [6] L. Chergui, Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity,, J. Evol. Equ., 9 (2009), 405. Google Scholar [7] R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations,, Nonlinear Anal., 53 (2000), 1017. Google Scholar [8] M. Grasselli and M. Pierre, Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems,, to appear., (). Google Scholar [9] A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities,, J. Differential Equations, 144 (1998), 313. Google Scholar [10] A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity,, Calc. Var. Partial Differential Equations, 9 (1999), 95. Google Scholar [11] A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity,, Asymptot. Ana., 26 (2001), 21. Google Scholar [12] A. Haraux and M. A. Jendoubi, The Łojasiewicz gradient inequality in the infinite dimensional Hilbert space framework,, J. Funct. Anal., 260 (2011), 2826. Google Scholar [13] A. Haraux, M. A. Jendoubi and O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations,, J. Evol. Equ., 3 (2003), 463. Google Scholar [14] S.-Z. Huang, "Gradient Inequalities,'', American Mathematical Society, (2006). Google Scholar [15] S. Z. Huang and P. Takac, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, Nonlinear Anal. Ser. A, 46 (2001), 675. Google Scholar [16] M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon,, J. Funct. Anal., 153 (1998), 187. Google Scholar [17] M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity,, J. Differential Equations, 144 (1998), 302. Google Scholar [18] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels,, Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, (1962), 87. Google Scholar [19] S. Łojasiewicz, "Ensembles semi-analytiques,", Preprint, (1965). Google Scholar [20] S. Łojasiewicz, Sur les trajectoires du gradient d'une fonction analytique,, Geometry seminars, (1984), 1982. Google Scholar [21] B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications,, Commun. Pure Appl. Anal., 9 (2010), 685. Google Scholar [22] J. Palis and W. de Melo, "Geometric Theory of Dynamical Systems. An Introduction,'', Springer Verlag, (1982). Google Scholar [23] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems,, Ann. of Math., 118 (1983), 525. Google Scholar

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##### References:
 [1] P.-A. Absil, R. Mahony and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions,, SIAM J. Optim., 16 (2005), 531. Google Scholar [2] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations,, Asymptotic Analysis, 69 (2010), 31. Google Scholar [3] I. Ben Hassen and L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation,, J. Dynam. Diff. Equa., 23 (2011), 315. Google Scholar [4] I. Ben Hassen and A. Haraux, Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy,, J. Funct. Anal., 260 (2011), 2933. Google Scholar [5] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity,, J. Dynam. Diff. Equa., 20 (2008), 643. Google Scholar [6] L. Chergui, Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity,, J. Evol. Equ., 9 (2009), 405. Google Scholar [7] R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations,, Nonlinear Anal., 53 (2000), 1017. Google Scholar [8] M. Grasselli and M. Pierre, Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems,, to appear., (). Google Scholar [9] A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities,, J. Differential Equations, 144 (1998), 313. Google Scholar [10] A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity,, Calc. Var. Partial Differential Equations, 9 (1999), 95. Google Scholar [11] A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity,, Asymptot. Ana., 26 (2001), 21. Google Scholar [12] A. Haraux and M. A. Jendoubi, The Łojasiewicz gradient inequality in the infinite dimensional Hilbert space framework,, J. Funct. Anal., 260 (2011), 2826. Google Scholar [13] A. Haraux, M. A. Jendoubi and O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations,, J. Evol. Equ., 3 (2003), 463. Google Scholar [14] S.-Z. Huang, "Gradient Inequalities,'', American Mathematical Society, (2006). Google Scholar [15] S. Z. Huang and P. Takac, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, Nonlinear Anal. Ser. A, 46 (2001), 675. Google Scholar [16] M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon,, J. Funct. Anal., 153 (1998), 187. Google Scholar [17] M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity,, J. Differential Equations, 144 (1998), 302. Google Scholar [18] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels,, Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, (1962), 87. Google Scholar [19] S. Łojasiewicz, "Ensembles semi-analytiques,", Preprint, (1965). Google Scholar [20] S. Łojasiewicz, Sur les trajectoires du gradient d'une fonction analytique,, Geometry seminars, (1984), 1982. Google Scholar [21] B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications,, Commun. Pure Appl. Anal., 9 (2010), 685. Google Scholar [22] J. Palis and W. de Melo, "Geometric Theory of Dynamical Systems. An Introduction,'', Springer Verlag, (1982). Google Scholar [23] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems,, Ann. of Math., 118 (1983), 525. Google Scholar
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