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 + cA 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.