On a second order dissipative ODE in Hilbert space with an integrable source term

Asymptotic behaviour of solutions is studied for some second order equations including the model case with γ > 0 and h ∈ L1(0, + ∞ H), Φ being continuouly differentiable with locally Lipschitz continuous gradient and bounded from below. In particular when Φ is convex, all solutions tend to minimize the potential Φ as time tends to infinity and the existence of one bounded trajectory implies the weak convergence of all solutions to equilibrium points.