APPLICATIONS OF THE ŁOJASIEWICZ–SIMON, GRADIENT INEQUALITY TO GRADIENT-LIKE EVOLUTION EQUATIONS

We prove convergence to equilibrium of global and bounded solutions of gradient-like evolution equations. Our abstract results are illustrated by several examples in finite and infinite dimensions.

[1]  M. Jendoubi,et al.  CONVERGENCE TO EQUILIBRIUM FOR SEMILINEAR PARABOLIC PROBLEMS IN , 2002 .

[2]  Piotr Rybka,et al.  Convergence of solutions to cahn-hilliard equation , 1999 .

[3]  Alain Haraux,et al.  Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity , 2001 .

[4]  H. Attouch,et al.  THE HEAVY BALL WITH FRICTION METHOD, I. THE CONTINUOUS DYNAMICAL SYSTEM: GLOBAL EXPLORATION OF THE LOCAL MINIMA OF A REAL-VALUED FUNCTION BY ASYMPTOTIC ANALYSIS OF A DISSIPATIVE DYNAMICAL SYSTEM , 2000 .

[5]  J. Bolte,et al.  A second-order gradient-like dissipative dynamical system with Hessian-driven damping.: Application to optimization and mechanics , 2002 .

[6]  G. F. Webb,et al.  Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces , 1979, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[7]  M. Jendoubi.  Convergence of Global and Bounded Solutions of the Wave Equation with Linear Dissipation and Analytic Nonlinearity , 1998 .

[8]  Ralph Chill,et al.  On the Łojasiewicz–Simon gradient inequality , 2003 .

[9]  G. P. Szegö,et al.  Stability theory of dynamical systems , 1970 .

[10]  S. Łojasiewicz.  Ensembles semi-analytiques , 1965 .

[11]  W. Ni,et al.  Locating the peaks of least energy solutions to a semilinear Neumann problem , 1993 .

[12]  Alain Haraux,et al.  Rate of decay to equilibrium in some semilinear parabolic equations , 2003 .

[13]  M. Kwong.  Uniqueness of positive solutions of Δu−u+up=0 in Rn , 1989 .

[14]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[15]  E. Feireisl,et al.  Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations , 1997, Differential and Integral Equations.

[16]  J. Palis,et al.  Geometric theory of dynamical systems : an introduction , 1984 .

[17]  A. Haraux,et al.  Systèmes dynamiques dissipatifs et applications , 1991 .

[18]  A. Haraux,et al.  Convergence of Solutions of Second-Order Gradient-Like Systems with Analytic Nonlinearities , 1998 .

[19]  John William Neuberger,et al.  Sobolev gradients and differential equations , 1997 .

[20]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[21]  J. Palis,et al.  Geometric theory of dynamical systems , 1982 .

[22]  Alain Haraux,et al.  Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity , 1999 .

[23]  M. Jendoubi.  A Remark on the Convergence of Global and Bounded Solutions for a Semilinear Wave Equation on ∝N , 2002 .

[24]  J. Ball.  On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations , 1978 .

[25]  P. Polácik,et al.  Nonconvergent Bounded Trajectories in Semilinear Heat Equations , 1996 .

[26]  Convergence of Global and Bounded Solutions of a Second Order Gradient like System with Nonlinear Dissipation and Analytic Nonlinearity , 2008 .

[27]  R. Chill,et al.  Convergence to Steady States of Solutions of Non-autonomous Heat Equations in $$\mathbb{R}^{N}$$ , 2007 .

[28]  Convergence to steady states of solutions of semilinear evolutionary integral equations , 2005 .

[29]  Constantine M. Dafermos,et al.  Asymptotic Behavior of Solutions of Evolution Equations , 1978 .

[30]  P. Lions.  On Positive Solutions of Semilinear Elliptic Equations in Unbounded Domains , 1988 .

[31]  Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity , 2009 .

[32]  Mohamed Ali Jendoubi,et al.  A Simple Unified Approach to Some Convergence Theorems of L. Simon , 1998 .

[33]  L. Simon.  Asymptotics for a class of non-linear evolution equations, with applications to geometric problems , 1983 .

[34]  E. Feireisl.  Long-time behavior and convergence for semilinear wave equations on ℝN , 1997 .