On the ultimate energy bound of solutions to some forced second order evolution equations with a general nonlinear damping operator

Under suitable growth and coercivity conditions on the nonlinear damping operator g which ensure non-resonance, we estimate the ultimate bound of the energy of the general solution to the equationüequation¨equationü(t) + Au(t) + g(˙ u(t)) = h(t), t ∈ R + , where A is a positive selfadjoint operator on a Hilbert space H and h is a bounded forcing term with values in H. In general the bound is of the form C(1 + ||h|| 4) where ||h|| stands for the L ∞ norm of h with values in H and the growth of g does not seem to play any role. If g behaves lie a power for large values of the velocity, the ultimate bound has a quadratic growth with respect to ||h|| and this result is optimal. If h is anti periodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs