Global existence of a heavy ball model with a varying nonnegative friction

In this paper, we study a dissipative dynamical system non linear of second order $ \ddot{x}(t)+\lambda(t)\, \dot{x}(t)+\nabla \Phi (x(t))=0,$ with the non-negative friction coefficient $\lambda \in \mathcal{C}([0,+\infty[)$ and the internal constraint $\Phi \in \mathcal{C}^2$.