Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

Abstract

<p style='text-indent:20px;'>In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.</p>

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

“…The PDE (89) can be seen as an Allen-Cahn equation with inertial term. Convergence to a steady state for space and/or time discretizations of (89) has been studied in [6,63,74,120] by means of a Lojasiewicz inequality (see also [79] for a nonlocal damping). Related PDEs have also been considered, such as the Cahn-Hilliard equation with inertial term [62,65] or the modified phase-field crystal equation [64,66].…”

Convergence to equilibrium for solutions of some forced discretized second-order gradient-like systems

“…Horsin and Jendoubi [13] established a general convergence result for sequences satisfying a specific angle condition in the case where ε = 1, α > 0 sufficiently small, and g n = 0.…”

“…It is easy to see that hypothesis (3) on F implies (see, for example, [13]) the following inequality:…”