A general class of second order differential equations

Abstract

In this short chapter we introduce some elementary tools of evolution equations which will be applied to the analysis of time domain potentials and integral operators. Most of these results in the abstract treatment of evolution equations can be obtained with elementary tools of the theory of strongly continuous semigroups of operators (of groups of isometries actually). An introduction to semigroup techniques applied to partial differential equations can be found in [58, Chapter 4] or [44], with a very general treatment given in [71]. The theory of semigroups of operators is rarely part of the mathematical toolbox for users of boundary integral equations. This is the reason why we include a self-contained approach in Appendix B using a simple point of view, namely the method of separation of variables.