On Hopf type functional derivative equations for □ u + cu + bu2 + au3 = 0 on [formula omitted]. I. Existence of solutions

Abstract

In this note, we construct solutions of certain Functional Derivative Equations (FDEs) of third order whose characteristics are represented by the non-linear Klein-Gordon (NLKG) equation on Ω × R where Ω is a bounded domain in R d ( d ⩽ 4). Moreover, we may solve a FDE which is considered as the many-time Hopf equation corresponding to the NLKG equation and which has the great resemblance (modulo integration by parts) to the Schwinger-Dyson (SD) equation of the Minkowski Φ 4 4 theory.