This monograph gives a systematic account of the theory of vector-valued Laplace transforms, ranging from representation theory to Tauberian theorems. In parallel, the theory of linear Cauchy problems and semigroups of operators is developed completely in the spirit of Laplace transforms. Existence and uniqueness, regularity, approximation and above all asymptotic behaviour of solutions are studied. Diverse applications to partial differential equations are given. The book contains an introduction to the Bochner integral and several appendices on background material. It is addressed to students and researchers interested in evolution equations, Laplace and Fourier transforms, and functional analysis. The second edition contains detailed notes on the developments in the last decade. They include, for instance, a new characterization of well-posedness of abstract wave equations in Hilbert space due to M. Crouzeix. Moreover new quantitative results on asymptotic behaviour of Laplace transforms have been added. The references are updated and some errors have been corrected.

Preface to the First Edition.- Preface to the Second Edition.- I Laplace Transforms and Well-Posedness of Cauchy Problems.- 1 The Laplace Integral.- 2 The Laplace Transform.- 3 Cauchy Problems.- II Tauberian Theorems and Cauchy Problems.- 4 Asymptotics of Laplace Transforms.- 5 Asymptotics of Solutions of Cauchy Problems.- III Applications and Examples.- 6 The Heat Equation.- 7 The Wave Equation.- 8 Translation Invariant Operators on Lp(Rn).- A Vector-valued Holomorphic Functions.- B Closed Operators.- C Ordered Banach Spaces.- D Banach Spaces which Contain c0.- E Distributions and Fourier Multipliers.- Bibliography.- Notation.- Index.