Integrated Semigroups and Related Partial Differential Equations

Abstract

We investigate the relationship between abstract linear evolution equations of heat, wave, and Schrodinger types in terms of well-posedness in Banach spaces. More precisely, we study our operators as generators of integrated semigroups and integrated cosine functions. As applications, we consider in a Banach space context p Ž N . an abstract Laplacian which generalizes the ordinary Laplacian ⌬ in the L ‫ޒ‬ -spaces. We obtain optimal results when compared to the classical situation where Ž . the generation results were obtained by M. Hieber Math. Ann. 291, 1991, 1᎐16 using completely different methods. Our main tools are the boundary value theorem for holomorphic semigroups and the abstract Weierstrass formula. Related partial differential equations have been considered previously by Bragg and Dettman. ᮊ 1997 Academic Press

Convoluted C-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines

“…In the rest of Section 5, we analyze relations between (local) integrated cosine functions as well as convoluted cosine functions with ultradistribution semigroups. Such results were first obtained by V. Keyantuo in [25,Theorem 3.1] and this theorem has been recently generalized and analyzed in [33,Theorem 4…”

“…We recall that V. Keyantuo proved in [25,Theorem 3.1] that if a densely defined operator A generates an exponentially bounded α-times integrated cosine function for some α 0 (this means that A is densely defined and that A generates an exponential distribution cosine function of [32]), then ±iA generate ultradistribution semigroups of [7]. Furthermore, the proof of [25, [33,Section 4].…”

“…We refer to [6][7][8][9][10][11][12][13][14] (see also [15,43]), [20,25,31,33,37,40] and [46] for further information concerning ultradistribution semigroups. We analyze in this paper ultradistribution and hyperfunction sines continuing the researches of H. Komatsu [31] and P.C.…”

Ultradistribution semigroups

“…The interested reader may consult the monographs [4][5][6][7] for a fairly complete information concerning the general theory of integrated semigroups, C-regularized semigroups, and convoluted semigroups. Ultradistribution semigroups with densely defined generators were analyzed by J. Chazarain in [8,9] (see also [10][11][12][13]), and the first results related to ultradistribution semigroups with nondensely defined generators were obtained by H. Komatsu in [14].…”

“…, finishing the proof of (a). The proofs of (b) and (c) follow from (a) and the argumentation given in the proofs of [13,Theorem 3.1] and [34,Theorem 4.3]. The proof of (d) is an immediate consequence of (a) and the fact that, for all c ∈ (0, 1), σ > 0, ς ∈ R, and a ∈ (0, 1 2 + c 2 ), there exist σ 1 > 0 and ς 1 ∈ R such that C \ {λ 2 : λ ∈ Π c,σ,ς } ⊆ −Π a,σ 1 ,ς 1 .…”