Bégout, Pascal, Bolte, Jérôme and Jendoubi, Mohamed Ali (2015) On damped second-order gradient systems. Journal of Differential Equations, 259 (7). pp. 3115-3143.

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Abstract

Using small deformations of the total energy, as introduced in \cite{MR1616968}, we establish that damped second order gradient systems
\begin{gather*}
u^\pp(t)+\gamma u^\p(t)+\nabla G(u(t))=0,
\end{gather*}
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 $\vphi(s)\ge c\sqrt s$ whenever the original function is definable and $C^2.$ 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.