Perturbations of Positive Semigroups with Applications is a
self-contained introduction to semigroup theory with emphasis on
positive semigroups on Banach lattices and perturbation techniques. The
first part of the book, which should be regarded as an extended
reference section, presents a survey of the results from functional
analysis, the theory of positive operators and the theory of semigroups
that are needed for the second, applied part of the book; worked
examples are provided to help absorb the theoretical material. The
second part then deals with the application of the developed theory to a
variety of problems ranging from the classical birth-and-death type
problems of population dynamics, through fragmentation models in both
conservative and mass loss regimes, to kinetic models.
Features of particular interest include:
- a survey of perturbation results for semigroups, beginning with the
classical results and ending with more recent developments such as the
Miyadera-Voigt perturbation theorem, Desch's results and an extension
of the Kato theorem to Kantorovic-Banach spaces;
- comprehensive analysis of the relations between the generator of a
semigroup and the conservativeness of this semigroup, providing
applicable techniques for characterization of the generator;
- a full description of the dynamics of birth-and-death problems;
- an analytical explanation of shattering and dust formation and of the
existence of multiple solutions for fragmentation equations;
- a detailed analysis of the linear Boltzmann equation with external
field with both classical and inelastic scattering kernels,
supplemented by an exhaustive treatment of boundary value problems for
a streaming operator;
- applications to the asymptotic analysis of kinetic equations.
The only prerequisites are a basic knowledge of functional analysis and
measure theory. This is an essential one-stop reference for graduate and
postgraduate students and researchers in applied analysis and
mathematical physics who are interested in the application of functional
analytic techniques to concrete problems arising in physics, biology and
engineering.
