Local Integrated Semigroups: Evolution with Jumps of Regularity

Singular evolution problems, regularization, and applications to physics, engineering, and biology

“…are special cases of K(t)-evolution operators; similarly for local n-i.s.g. [2], see [30]; also a similar situation holds for C-semigroups.…”

“…We now describe in the general context of Section 1 but considering only scalar kernels, a very general generation theorem which contains not only the recent one in [2] but earlier results of Beals [3], [4], Chazarain [5], [6], Emamirad [15], Cioranescu and Zsidó [10], Cioranescu [7]; this and other related results are in [8], [9].…”

“…n-i.s.g. in terms of the growth of (the norm of) the resolvent R(z, A) in some subdomain of {z ∈ C : Re z > 0}, an "exponential domain" indeed (see [2]) ( 17 ). On the other hand, loc.…”

Distribution semigroups and abstract Cauchy problems

“…Moreover, the local strongly continuous representations (see section 4) of the distribution semigroups thus obtained are exactly the local integrated semigroups (for the distribution semigroups in the sense of [16] a relation to local integrated semigroups has been obtained in [23], Theorem 5.6; see also [15], Theorem 7.6 or [2], Theorem 7.2). So the theory presented here can be viewed as a unified approach to treating Cauchy problems via fundamental solutions, distribution semigroups or local integrated semigroups-without any denseness assumptions.…”

“…The author wants to thank V. Wrobel for many helpful discussions, L. Weis for preprints of [2] and [10] and V. Keyantuo for a copy of [15].…”