Well-posedness to the discrete collision-induced breakage equation and various properties of solutions

TL;DR

This paper establishes the existence, uniqueness, and qualitative properties of solutions to a discrete collision-induced breakage equation, providing a rigorous mathematical foundation for understanding such particle systems.

Contribution

It proves the existence and uniqueness of solutions for a broad class of collision kernels and analyzes their long-term behavior and properties.

Findings

Existence of mass-conserving solutions for all times.

Uniqueness of solutions under general conditions.

Analysis of moments, differentiability, and large-time behavior.

Abstract

A discrete version of the nonlinear collision-induced breakage equation is studied. Existence of solutions is investigated for a broad class of unbounded collision kernels and daughter distribution functions, the collision kernel $a_{i,j}$ satisfiying $a_{i,j} \leq A i j$ for some $A>0$. More precisely, it is proved that given suitable conditions, there exists at least one mass-conserving solution for all times. A result on the uniqueness of solutions is also demonstrated under reasonably general conditions. Furthermore, the propagation of moments, differentiability, and the continuous dependence of solutions are established, along with some invariance properties and the large-time behaviour of solutions.