Analyticity and regularity for a class of second order evolution equations

The regularity conservation as well as the smoothing effect are studied for the equation u″+Au+cA^αu′ = 0, where A is a positive selfadjoint operator on a real Hilbert space H and α ∈ (0, 1]; c > 0. When α ≥ 1/2 the equation generates an analytic semigroup on D(A^1/2) × H, and if α ∈ (0, 1/2) a weaker optimal smoothing property is established. Some conservation prop- erties in other norms are also established, as a typical example, the strongly dissipative wave equation u_tt – Δu – cΔu_t = 0 with Dirichlet boundary conditions in a bounded domain is given, for which the space C₀(Ω) × C₀(Ω) is conserved for t > 0, which presents a sharp contrast with the conservative case u_tt – Δu = 0 for which C0()-regularity can be lost even starting from an initial state (u₀, 0) with u₀ 2 C₀(Ω) ⋂ C¹(Ω).