Asymptotics for a Dissipative Dynamical System with Linear and Gradient-Driven Damping

We study, in the setting of a real Hilbert space H, the asymptotic behavior of trajectories of the second-order dissipative dynamical system with linear and gradient-driven nonlinear dampingwhere λ > 0 and f, Φ: H → R are two convex differentiable functions. It is proved that if Φ is coercive and bounded from below, then the trajectory converges weakly towards a minimizer of Φ. In particular, we state that under suitable conditions, the trajectory strongly converges to the minimizer of Φ exponentially or polynomially.