Quasi-Distribution Semigroups and Integrated Semigroups

Abstract

Suppose X is a Banach space. We introduce the concept of quasi-distribution semigroups on X, as a generalization of the concept of distribution semigroups introduced by Lions in [19]. In our approach, the generator A of a quasi-distribution semigroup may not be densely defined. Also introduced is a functional calculus for A in terms of the Fourier transform. For fixed r>0, k # N _ [0], define order (r, k) for a quasi-distribution semigroup and define an F r, k functional calculus. We prove that A generates a (k+1)-times integrated semigroup of exponential type r with local Lipschitz continuity if and only if A generates a quasi-distribution semigroup of order (r, k) if and only if A has an F r, k functional calculus. When A is densely defined, a quasi-distribution semigroup reduces to a distribution semigroup in the sense of Lions.1997 Academic Press

Local and global solutions of well-posed integrated Cauchy problems

“…Particular classes of distribution semigroups have since been considered; for example quasi-distribution semigroups were introduced and studied by Wang in [21]. A brief description of distribution semigroups as well as references on the subject can be found in [1] and [17].…”

“…has been studied in detail for α ∈ N ( [3], [21]) and later for α ∈ R + ( [15]). The Cauchy problem C α (κ) is well-posed if for all x ∈ X there exists a unique solution of C α (κ).…”

“…for κ > t + s ≥ t, s ≥ 0, and x ∈ X (nondegenerate in the usual sense: if S α (t)x = 0 for all t ∈ [0, κ) then x = 0); see [2], [15], [21]. It is straightforward to check that (S β (t)) t∈[0,κ) defined by…”

“…Quasi-distribution semigroups and global solutions of the abstract Cauchy problem. In this section we start considering quasidistribution semigroups introduced in [21]. …”

Distribution semigroups and abstract Cauchy problems

“…Now by the resolvent equation After this paper had been accepted the author learned that the definition of a DSG, i.e. (3) and (4), as well as the assertions of Corollary 4.8 and Theorem 4.11 also appear in [25].…”