# American Institute of Mathematical Sciences

November  2012, 11(6): 2393-2416. doi: 10.3934/cpaa.2012.11.2393

## Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems

 1 Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, I-20133 Milano 2 Laboratoire de Mathématiques et Applications UMR CNRS 6086, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil

Received  October 2010 Revised  December 2010 Published  April 2012

Following a result of Chill and Jendoubi in the continuous case, we study the asymptotic behavior of sequences $(U^n)_n$ in $R^d$ which satisfy the following backward Euler scheme:

$\varepsilon\frac{(U^{n+1}-2U^n+U^{n-1}}{\Delta t^2} +\frac{U^{n+1}-U^n}{\Delta t}+\nabla F(U^{n+1})=G^{n+1}, n\ge 0,$

where $\Delta t>0$ is the time step, $\varepsilon\ge 0$, $(G^{n+1})_n$ is a sequence in $R^d$ which converges to $0$ in a suitable way, and $F\in C^{1,1}_{l o c}(R^d, R)$ is a function which satisfies a Łojasiewicz inequality. We prove that the above scheme is Lyapunov stable and that any bounded sequence $(U^n)_n$ which complies with it converges to a critical point of $F$ as $n$ tends to $\infty$. We also obtain convergence rates. We assume that $F$ is semiconvex for some constant $c_F\ge 0$ and that $1/\Delta t Citation: Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393 ##### 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. doi: 10.1137/040605266. [2] H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features,, Math. Program., 116 (2009), 5. [3] H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and {PDE's},, J. Convex Anal., 15 (2008), 485. [4] T. Bárta, R. Chill and E. Fašangová, Every ordinary differential equation with a strict Lyapunov function is a gradient system,, submitted., (). doi: 10.1007/s00605-011-0322-4. [5] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations,, Asymptot. Anal., 69 (2010), 31. [6] P. Bénilan, M. G. Crandall and A. Pazy, "Bonnes solutions'' d'un problème d'évolution semi-linéaire,, C. R. Acad. Sci. Paris S\'er. I Math., 306 (1988), 527. [7] J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity,, Trans. Amer. Math. Soc., 362 (2010), 3319. [8] H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'', North-Holland Publishing Co., (1973). [9] J. V. Chaparova, L. A. Peletier and S. A. Tersian, Existence and nonexistence of nontrivial solutions of semilinear fourth- and sixth-order differential equations,, Adv. Differential Equations, 8 (2003), 1237. [10] R. Chill and A. Haraux and M. A. Jendoubi, Applications of the Łojasiewicz-Simon gradient inequality to gradient-like evolution equations,, Anal. Appl. (Singap.), 7 (2009), 351. [11] R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations,, Nonlinear Anal., 53 (2003), 1017. [12] K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals,, Phys. Rev. E, 70 (2004). [13] K. R. Elder, N. Provatas, J. Berry, P. Stefanovic and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing,, Phys. Rev. B, 75 (2007). [14] H. Gajewski and J. A. Griepentrog, A descent method for the free energy of multicomponent systems,, Discrete Contin. Dyn. Syst., 15 (2006), 505. [15] P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.051110. [16] M. Grasselli, H. Petzeltová and G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system,, Commun. Pure Appl. Anal., 5 (2006), 827. [17] M. Grasselli, H. Wu and S. Zheng, Asymptotic behavior of a nonisothermal Ginzburg-Landau model,, Quart. Appl. Math., 66 (2008), 743. [18] A. Haraux, "Syst\emes dynamiques dissipatifs et applications,'', Masson, (1991). [19] A. Haraux, Slow and fast decay of solutions to some second order evolution equations,, J. Anal. Math., 95 (2005), 297. doi: 10.1007/BF02791505. [20] A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities,, J. Differential Equations, 144 (1998), 313. doi: 10.1006/jdeq.1997.3393. [21] 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. doi: 10.1007/s005260050133. [22] S.-Z. Huang, "Gradient Inequalities,'', American Mathematical Society, (2006). [23] S.-Z. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, Nonlinear Anal., 46 (2001), 675. doi: 10.1016/S0362-546X(00)00145-0. [24] M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon,, J. Funct. Anal., 153 (1998), 187. doi: 10.1006/jfan.1997.3174. [25] 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. doi: 10.1006/jdeq.1997.3392. [26] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels,, in, (1962). [27] S. Łojasiewicz, "Ensembles semi-analytiques,", I.H.E.S. Notes, (1965). [28] P.-E. Maingé, Asymptotic convergence of an inertial proximal method for unconstrained quasiconvex minimization,, J. Global Optim., 45 (2009), 631. doi: 10.1007/s10898-008-9388-5. [29] B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications,, Commun. Pure Appl. Anal., 9 (2010), 685. doi: 10.3934/cpaa.2010.9.685. [30] M. Polat, Global attractor for a modified Swift-Hohenberg equation,, Comput. Math. Appl., 57 (2009), 62. doi: 10.1016/j.camwa.2008.09.028. [31] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems,, Ann. of Math., 118 (1983), 525. doi: 10.2307/2006981. [32] L. Song, Y. Zhang and T. Ma, Global attractor of a modified Swift-Hohenberg equation in$H^k$spaces,, Nonlinear Anal., 72 (2010), 183. doi: 10.1016/j.na.2009.06.103. [33] A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,'', Cambridge University Press, (1996). [34] J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Phys. Rev. A, 15 (1977), 319. doi: 10.1103/PhysRevA.15.319. [35] S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal., 3 (2004), 921. doi: 10.3934/cpaa.2004.3.921. [36] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities,, Discrete Contin. Dyn. Syst., 11 (2004), 351. doi: 10.3934/dcds.2004.11.351. show all references ##### 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. doi: 10.1137/040605266. [2] H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features,, Math. Program., 116 (2009), 5. [3] H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and {PDE's},, J. Convex Anal., 15 (2008), 485. [4] T. Bárta, R. Chill and E. Fašangová, Every ordinary differential equation with a strict Lyapunov function is a gradient system,, submitted., (). doi: 10.1007/s00605-011-0322-4. [5] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations,, Asymptot. Anal., 69 (2010), 31. [6] P. Bénilan, M. G. Crandall and A. Pazy, "Bonnes solutions'' d'un problème d'évolution semi-linéaire,, C. R. Acad. Sci. Paris S\'er. I Math., 306 (1988), 527. [7] J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity,, Trans. Amer. Math. Soc., 362 (2010), 3319. [8] H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'', North-Holland Publishing Co., (1973). [9] J. V. Chaparova, L. A. Peletier and S. A. Tersian, Existence and nonexistence of nontrivial solutions of semilinear fourth- and sixth-order differential equations,, Adv. Differential Equations, 8 (2003), 1237. [10] R. Chill and A. Haraux and M. A. Jendoubi, Applications of the Łojasiewicz-Simon gradient inequality to gradient-like evolution equations,, Anal. Appl. (Singap.), 7 (2009), 351. [11] R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations,, Nonlinear Anal., 53 (2003), 1017. [12] K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals,, Phys. Rev. E, 70 (2004). [13] K. R. Elder, N. Provatas, J. Berry, P. Stefanovic and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing,, Phys. Rev. B, 75 (2007). [14] H. Gajewski and J. A. Griepentrog, A descent method for the free energy of multicomponent systems,, Discrete Contin. Dyn. Syst., 15 (2006), 505. [15] P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.051110. [16] M. Grasselli, H. Petzeltová and G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system,, Commun. Pure Appl. Anal., 5 (2006), 827. [17] M. Grasselli, H. Wu and S. Zheng, Asymptotic behavior of a nonisothermal Ginzburg-Landau model,, Quart. Appl. Math., 66 (2008), 743. [18] A. Haraux, "Syst\emes dynamiques dissipatifs et applications,'', Masson, (1991). [19] A. Haraux, Slow and fast decay of solutions to some second order evolution equations,, J. Anal. Math., 95 (2005), 297. doi: 10.1007/BF02791505. [20] A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities,, J. Differential Equations, 144 (1998), 313. doi: 10.1006/jdeq.1997.3393. [21] 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. doi: 10.1007/s005260050133. [22] S.-Z. Huang, "Gradient Inequalities,'', American Mathematical Society, (2006). [23] S.-Z. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, Nonlinear Anal., 46 (2001), 675. doi: 10.1016/S0362-546X(00)00145-0. [24] M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon,, J. Funct. Anal., 153 (1998), 187. doi: 10.1006/jfan.1997.3174. [25] 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. doi: 10.1006/jdeq.1997.3392. [26] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels,, in, (1962). [27] S. Łojasiewicz, "Ensembles semi-analytiques,", I.H.E.S. Notes, (1965). [28] P.-E. Maingé, Asymptotic convergence of an inertial proximal method for unconstrained quasiconvex minimization,, J. Global Optim., 45 (2009), 631. doi: 10.1007/s10898-008-9388-5. [29] B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications,, Commun. Pure Appl. Anal., 9 (2010), 685. doi: 10.3934/cpaa.2010.9.685. [30] M. Polat, Global attractor for a modified Swift-Hohenberg equation,, Comput. Math. Appl., 57 (2009), 62. doi: 10.1016/j.camwa.2008.09.028. [31] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems,, Ann. of Math., 118 (1983), 525. doi: 10.2307/2006981. [32] L. Song, Y. Zhang and T. Ma, Global attractor of a modified Swift-Hohenberg equation in$H^k$spaces,, Nonlinear Anal., 72 (2010), 183. doi: 10.1016/j.na.2009.06.103. [33] A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,'', Cambridge University Press, (1996). [34] J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Phys. Rev. A, 15 (1977), 319. doi: 10.1103/PhysRevA.15.319. [35] S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal., 3 (2004), 921. doi: 10.3934/cpaa.2004.3.921. [36] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities,, Discrete Contin. Dyn. Syst., 11 (2004), 351. doi: 10.3934/dcds.2004.11.351.  [1] 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 [2] Peter Takáč. Stabilization of positive solutions for analytic gradient-like systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 947-973. doi: 10.3934/dcds.2000.6.947 [3] Zhuchun Li, Yi Liu, Xiaoping Xue. Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 345-367. doi: 10.3934/dcds.2019014 [4] Xuewei Ju, Desheng Li, Jinqiao Duan. 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