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 $\ddot{x}(t)+\lambda (t)\dot{x}(t)+\mathrm{\nabla }\mathrm{\Phi }(x(t))=0,$ with the non-negative friction coefficient $\lambda \in \mathcal{C}([0,+\mathrm{\infty }[)$ and the internal constraint $\mathrm{\Phi }\in {\mathcal{C}}^2$.