Integrated Semigroups and Related Partial Differential Equations

We investigate the relationship between abstract linear evolution equations of heat, wave, and Schr¨odinger 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 an abstract Laplacian which generalizes the ordinary Laplacian D in the L p RN .- spaces. We obtain optimal results when compared to the classical situation where 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.