4 General Properties of Solutions of Classical Field Equations

Abstract

The first basic question is to identify the possible configurations of the systems (3.1), namely the set X of initial data for which the time evolution is well defined and which is mapped onto itself by time evolution. In the mathematical language, one has to find the functional space X for which the Cauchy problem is well posed. In order to see this, one has to give conditions on U′ (ϕ ) and to specify the class of initial data or, equivalently, the class of solutions one is interested in. Here one faces an apparently technical mathematical problem, which has also deep physical connections.