Global solution to the Cauchy problem on a universe fireworks model

TL;DR

This paper establishes the existence, uniqueness, regularity, and asymptotic behavior of solutions to a Boltzmann-like universe fireworks model, providing a theoretical framework for understanding such complex systems.

Contribution

It introduces a novel approach to proving existence and uniqueness for a universe fireworks model using contractive mappings in Banach spaces, inspired by DiPerna & Lions' methods.

Findings

Proves global existence and uniqueness of solutions.

Analyzes regularity and long-term behavior of solutions.

Provides a mathematical framework for universe fireworks models.

Abstract

We prove existence and uniqueness of the global solution to the Cauchy problem on a universe fireworks model with finite total mass at the initial state when the ratio of the mass surviving the explosion, the probability of the explosion of fragments and the probability function of the velocity change of a surviving particle satisfy the corresponding physical conditions. Although the nonrelativistic Boltzmann-like equation modeling the universe fireworks is mathematically easy, this paper leads rather theoretically to an understanding of how to construct contractive mappings in a Banach space for the proof of the existence and uniqueness by means of methods taken from the famous work by DiPerna & Lions about the Boltzmann equation. We also show both the regularity and the time-asymptotic behavior of solution to the Cauchy problem.