The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions

TL;DR

This paper analyzes the stationary solutions of the fragmentation equation with size diffusion, revealing their asymptotic behaviors at small and large sizes, and providing explicit solutions for specific cases.

Contribution

It characterizes the asymptotic behaviors of stationary solutions at both small and large sizes, highlighting the influence of fragmentation rates and distributions, and offers explicit solutions for certain coefficients.

Findings

Large size solutions behave like stretched exponentials influenced by fragmentation rate at infinity.

Small size solutions are at most linear, with possible non-algebraic behavior.

Explicit solutions are derived for specific fragmentation coefficients.

Abstract

The small and large size behavior of stationary solutions to the fragmentation equation with size diffusion is investigated. It is shown that these solutions behave like stretched exponentials for large sizes, the exponent in the exponential being solely given by the behavior of the overall fragmentation rate at infinity. In contrast, the small size behavior is partially governed by the daughter fragmentation distribution and is at most linear, with possibly non-algebraic behavior. Explicit solutions are also provided for particular fragmentation coefficients.