On the modified entropy equation

TL;DR

This paper investigates the solutions of a modified entropy equation on the positive cone in real space, exploring its connections to the classical entropy equation and identifying regular solutions.

Contribution

It introduces and solves a modified entropy equation, characterizes its regular solutions, and examines its relationship with the classical entropy equation.

Findings

Derived explicit solutions for the modified entropy equation.

Established conditions for regular solutions.

Explored links between the modified and classical entropy equations.

Abstract

The object of this paper is to solve the so--called modified entropy equation \[ f<x, y, z>=f<x, y+z, \mathbf{0}>+ \mu<y+z>f<\mathbf{0}, \frac{y}{y+z}, \frac{z}{y+z}>, \] on the positive cone of $\R^{k}$, where $\mu$ is a given multiplicative function on this cone. After that the regular solutions of this equation are determined. Furthermore we investigate its connection between the entropy equation and other equations, as well.