On the Minimal Extension ofC0-Semigroups for Second-Order Damped Equations

Abstract

The present paper deals with a minimal extension of the classical semigroup theory for second-order damped differential equations in Banach spaces with closed, densely defined linear operators as coefficients. We do not ask anymore from our operators than in the case of first-order equations, i.e., semigroups. We present here generalizations of the Miyadera–Phillips–Feller theorem, the Hille type theorem, and the Trotter–Kato type theorem. The method is quite general and could be used for equations of any order. We focus our attention on a particular dynamical operator solution or main propagator and we assume some properties about it. From this we can obtain some information about the complementary basis operator solutions or secondary propagators.