In a Hilbert space setting H , for convex optimization, we analyse the fast convergence properties as t → + ∞ of the trajectories t ↦ u ( t ) ∈ H generated by a third-order in time evolution system. The function f : H → R to minimize is supposed to be convex, continuously differentiable, with argmin H f ≠ ∅ . It enters into the dynamic through its gradient. Based on this new dynamical system, we improve the results obtained by Attouch et al. [Fast convex optimization via a third-order in time evolution equation. Optimization. 2020;71(5):1275–1304]. As a main result, when the damping parameter α satisfies α > 3 , we show that f ( u ( t ) ) − inf H f = o ( 1 / t 3 ) as t → + ∞ , as well as the convergence of the trajectories. We complement these results by introducing into the dynamic an Hessian-driven damping term, which reduces the oscillations. In the case of a strongly convex function f, we show an autonomous evolution system of the third-order in time with an exponential rate of convergence. All these results have natural extensions to the case of a convex lower semicontinuous function f : H → R ∪ { + ∞ } . Just replace f with its Moreau envelope.