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 nonresonance, we estimate the ultimate bound of the energy of the general solution to the equation u spexpressioncing diexpressioneresis (t) + Au(t)+g( ?u(t)) = h(t), t is an element of R+, where A is a positive self ad joint 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 + IIhII(4)), where IIh II stands for the L-infinity norm of h with values in H and the growth of g does not seem to play any role. If g behaves like a power for large values of the velocity, the ultimate bound has quadratic growth with respect to IIh II and this result is optimal. If h is antiperiodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.