Stability of selfsimilar solutions to the fragmentation equation with polynomial daughter fragments distribution

TL;DR

This paper analyzes the stability and regularity of self-similar solutions to the fragmentation equation with polynomial daughter distribution, demonstrating exponential decay and smoothness under various perturbations.

Contribution

It provides a detailed stability analysis of self-similar solutions with explicit Laguerre polynomial expansions, including cases with power-law decay and real analytic distributions.

Findings

Self-similar solutions exhibit exponential decay and $C^{ abla}$ regularity.

Stability under smooth exponentially decaying perturbations is established.

Results include stability for perturbations with power-law decay at infinity.

Abstract

We study fragmentation equations with power-law fragmentation rates and polynomial daughter fragments distribution function $p(s)$. The corresponding selfsimillar solutions are analysed and their exponentially decaying asymptotic behaviour and $C^{\infty }$ regularity deduced. Stability of selfsimilar solutions (under smooth exponentially decaying perturbations), with sharp exponential decay rates in time are proved, as well as $C^{\infty }$ regularity of solutions for $t>0$. The results are based on explicit expansion in terms of generalized Laguerre polynomials and the analysis of such expansions. For perturbations with power-law decay at infinity stability is also proved. Finally, we consider real analytic $p(s)$.