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(Communicated by M. Paˇ sic) Abstract. By using differential inequalities, an essentially optimal L ∞ (R,D(A)) bound of the unique bounded solution of u  +cAu  +A 2 u = f(t) is obtained whenever A = A ∗ λI is a bounded or unbounded linear operator on a real Hilbert space H and λ,c are positive constants, while f ∈ L ∞ (R,H).

Sharp estimates of bounded solutions to a second-order forced equation with structural damping