Vector-valued Laplace Transforms and Cauchy Problems

The book "Vector-valued Laplace Transforms and Cauchy Problems" offers a comprehensive and systematic presentation of the theory of vector-valued Laplace transforms. It addresses both representation theory and Tauberian theorems, while simultaneously developing the theory of linear Cauchy problems and semigroups of operators in the context of Laplace transforms. The investigation includes aspects such as existence and uniqueness, regularity, approximation, and particularly the asymptotic behavior of solutions. Additionally, it presents various applications to partial differential equations. The book includes an introduction to the Bochner integral as well as several appendices that provide background material. It is aimed at students and researchers interested in evolution equations, Laplace and Fourier transforms, and functional analysis. The second edition offers detailed notes on developments from the last decade, including new characterizations of the well-posedness of abstract wave equations in Hilbert space and updated references.