Let A be a positive self-adjoint linear operator acting on a real Hilbert space H and α, c be positive constants. We show that all solutions of the evolution equation u" + Au + c A^α u' = 0 with u(0) ∈ D( A_1/2), u (0) ∈ H belong for all t > 0 to the Gevrey space G(A, σ) with σ = min{ 1/ α , 1 /1−α }. This result is optimal in the sense that σ can not be reduced in general. For the damped wave equation (SDW)_α corresponding to the case where A = −∆ with domain D(A) = {w ∈ H^1_0 (Ω), ∆w ∈ L^2 (Ω)} with Ω any open subset of R^N and (u(0), u (0)) ∈ H^1_ 0 (Ω)×L^2 (Ω), the unique solution u of (SDW)_α satisfies ∀t > 0, u(t) ∈ G^s (Ω) with s = min{ 1 2α , 1 2(1−α) }, and this result is also optimal. Mathematics Subject Classification 2010 (MSC2010): 35L10, 35B65, 47A60.