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(alpha)u' = 0, where A is a positive selfadjoint operator on a real Hilbert space H and alpha is an element of (0, 1]; c > 0. When alpha >= 1/2 the equation generates an analytic semigroup on D(A(1/2)) x H, and if alpha is an element of (0,1/2) a weaker optimal smoothing property is established. Some conservation properties in other norms are also established, as a typical example, the strongly dissipative wave equation u(tt) - Delta(u) - c Delta u(t) = 0 with Dirichlet boundary conditions in a bounded domain is given, for which the space C-0(Omega) x C-0(Omega) is conserved for t > 0, which presents a sharp contrast with the conservative case u(tt) - Delta u = 0 for which C-0(Omega)-regularity can be lost even starting from an initial state (u(0), 0) with u(0) is an element of C-0(Omega) boolean AND C-1((Omega) over bar).