A class of C-distribution semigroups unifying the class of (quasi-) distribution semigroups of Wang and Kunstmann (when C = I) is introduced. Relations between C-distribution semigroups and integrated C-semigroups are given. Dense C-distribution semigroups as well as weak solutions of the corresponding Cauchy problems are also considered.
We introduce various classes of distribution semigroups distinguished by their behavior at the origin. We relate them to quasi-distribution semigroups and integrated semigroups. A class of such semigroups, called strong distribution semigroups, is characterized through the value at the origin in the sense of Łojasiewicz. It contains smooth distribution semigroups as a subclass. Moreover, the analysis of the behavior at the origin involves intrinsic structural results for semigroups. To this purpose,...
We study a classification of κ-times integrated semigroups (for κ > 0) by their (uniform) rate of convergence at the origin: as t → 0 (0 ≤ α ≤ κ). By an improved generation theorem we characterize this behaviour by Hille-Yosida type estimates. Then we consider integrated semigroups with holomorphic extension and characterize their convergence at the origin, as well as the existence of boundary values, by estimates of the associated holomorphic semigroup. Various examples illustrate these results....