WELL-POSEDNESS OF ABSTRACT CAUCHY-PROBLEMS FOR 2ND-ORDER DIFFERENTIAL-EQUATIONS

This paper concerns the abstract Cauchy problem (ACP) for an evolution equation of second order in time. Let A be a closed linear operator with domain D(A) dense in a Banach space X. We first characterize the exponential wellposedness of ACP on D(A k+1), k te N. Next let {C(t);t te R} be a family of generalized solution operators, on [D(A k)] to X, associated with an exponentially wellposed ACP on D(A k+1). Then we define a new family {T(t); Re t>0} by the abstract Weierstrass formula. We show that {T(t)} forms a holomorphic semigroup of class (H k) on X. © 1990 The Weizmann Science Press of Israel.