Asymptotics for some nonlinear O.D.E. of the second order

Using small deformations of the total energy, as introduced in [31], we establish that damped second order gradient systems 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 $\phi (s)⩾c\sqrt{s}$ 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.

In this paper the initial value problem and global properties of solutions are studied for the scalar second order ODE: ${\left({\left|u^{\prime }\right|}^l{u}^{\prime }\right)}^{\prime }+c{\left|u^{\prime }\right|}^{\alpha }{u}^{\prime }+d{\left|u\right|}^{\beta }u=0$, where $\alpha ,\beta ,l,c,d$ are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated.

We consider the second-order gradient-like system$\text{U}{}_{\mathrm{tt}}\text{+g(U}{}_{\mathrm{t}}\text{)=∇F(U)}\text{(1)}$whereF: $\text{R}$^N→$\text{R}$ is analytic andg: $\text{R}$^N→$\text{R}$^Nis Lipschitz and coercive withg(0)=0.  We prove the convergence of global and bounded solutions of (1) to some equilibrium points.