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.