Asymptotics for some semilinear hyperbolic equations with non-autonomous damping

Asymptotic for a second-order evolution equation with convex potential andvanishing damping term

“…Proof of Theorem 1.2. We will use a modified version of a method introduced by Cabot et Frankel in [2] and recently developed in [3]. Let x * ∈ arg min Φ and define the function h :…”

Asymptotic for the perturbed heavy ball system with vanishing damping term

“…In [6], Haraux and Jendoubi extended the weak convergence result of Alvarez to the case where the source term g belongs to the space L 1 (0, +∞; H). Recently, Cabot and Frankel [5] studied (1.1) where g = 0 and α ∈]0, 1[. They proved that every bounded solution x(t) (i.e.…”

Sequential formulae for the normal cone to sublevel sets

“…When +∞ 0 γ(t) dt = +∞ and argmin Φ = ∅, it can be easily proved that lim t→+∞ Φ(x(t)) = min Φ; see [15,18] where energy-like arguments are used. We will assume in the sequel that there exists a solution x ∈ W 2,1 loc (R + , R N ) satisfying (E) almost everywhere.…”

“…The case of a constant coefficient γ > 0 gives rise to the so-called "Heavy Ball with Friction" system. See also [18] for a related study in the framework of semilinear hyperbolic equations. The case of a vanishing damping parameter γ(t) → 0 has been recently investigated in [15,16] for a function Φ of class C 1 .…”