Semigroups and Evolutionary Equations

TL;DR

This paper explores how strongly continuous semigroups can be linked to evolutionary equations by defining appropriate function spaces and using extrapolation spaces, with applications to infinite-dimensional differential-algebraic and delay equations.

Contribution

It introduces a framework for associating semigroups with evolutionary equations, including the treatment of initial value problems via extrapolation spaces.

Findings

Semigroups can be associated with infinite-dimensional differential-algebraic equations.

The framework applies to hyperbolic delay equations.

Initial value problems are effectively formulated within this semigroup approach.

Abstract

We show how strongly continuous semigroups can be associated with evolutionary equations. For doing so, we need to define the space of admissible history functions and initial states. Moreover, the initial value problem has to be formulated within the framework of evolutionary equations, which is done by using the theory of extrapolation spaces. The results are applied to two examples. First, differential-algebraic equations in infinite dimensions are treated and it is shown, how a C_{0}-semigroup can be associated with such problems. In the second example we treat a concrete hyperbolic delay equation.