2014/11/30 by Mohamed Ali Jendoubi, Pascal Bégout, Jérôme Bolte +1 · 1 citation
Mathematics · #math.AP
paper · pdf · doi:10.1016/j.jde.2015.04.016
published as Journal of Differential Equations, Elsevier, 2015, 259 (7 \& 8), pp.3115-3143 \& 4412
arxiv created 2018/11/22 · arxiv updated 2019/04/21
Using small deformations of the total energy, as introduced in [31], we establish that damped second order gradient systems u′′(t)+γu^′(t)+∇ G(u(t))=0,may be viewed as quasi-gradient systems. In order to study the asymptotic behavior of these systems, we prove that any (nontrivial) desingularizing function appearing in KL inequality satisfies φ(s)≥ c√ s whenever the original function is definable and C2. Variants to this result are given. These facts are used in turn to prove that a desingularizing function of the potential G also desingularizes the total energy and its deformed versions. Our approach brings forward several results interesting for their own sake: we provide an asymptotic alternative for quasi-gradient systems, either a trajectory converges, or its norm tends to infinity. The convergence rates are also analyzed by an original method based on a one-dimensional worst-case gradient system.We conclude by establishing the convergence of solutions of damped second order systems in various cases including the definable case. The real-analytic case is recovered and some results concerning convex functions are also derived.