Semigroup Methods for Fragmentation Models
Jacek Banasiak, Wilson Lamb, Philippe Laurençot · 2019
An important role in the analysis of the long-term behaviour of semigroups is played by the peripheral spectrum. Crucial for the understanding of applications of functional analytic methods, such as semigroup theory, to concrete equations is the realisation that there is no one-to-one mapping between the model and its abstract formulation. Another important property of fractional powers of generators, and of the corresponding analytic semigroups, which will used in the sequel. The solution semigroup must be a semigroup of positive operators. A strongly continuous semigroup is positive if and only if its generator is resolvent positive. If the mass does not change, then the semigroup describing the evolution is conservative for positive initial data and is called a stochastic semigroup.