Abstract formulation of the Cole-Hopf transform

TL;DR

This paper presents an abstract operator-based formulation of the Cole-Hopf transform, generalizing it to Banach spaces and linking it to nonlinear semigroup theory through a relativistic evolution equation framework.

Contribution

It introduces a novel operator representation of the Cole-Hopf transform using logarithmic representation of generators, applicable in any spatial dimension.

Findings

Operator representation of Cole-Hopf transform derived

Generalization to Banach spaces achieved

Connection to nonlinear semigroup theory established

Abstract

Operator representation of Cole-Hopf transform is obtained based on the logarithmic representation of infinitesimal generators. For this purpose the relativistic formulation of abstract evolution equation is introduced. Even independent of the spatial dimension, the Cole-Hopf transform is generalized to a transform between linear and nonlinear equations defined in Banach spaces. In conclusion a role of transform between the evolution operator and its infinitesimal generator is understood in the context of generating nonlinear semigroup.