On Global Asymptotic Stability for the diffusive Carr-Penrose Model

TL;DR

This paper investigates the long-term behavior of a diffusive version of the Carr-Penrose model, demonstrating that solutions tend toward the eta=1 self-similar solution, supporting its physical relevance.

Contribution

It proves that solutions with compact support converge to the eta=1 self-similar solution in the diffusive Carr-Penrose model.

Findings

Solutions with compact support tend to the eta=1 self-similar solution over time.

Diffusion acts as a mechanism selecting the physically relevant self-similar solution.

Supports the idea that eta=1 solution is the unique long-term attractor.

Abstract

This paper is concerned with large time behavior of the solution to a diffusive perturbation of the linear LSW model introduced by Carr and Penrose. Like the LSW model, the Carr-Penrose model has a family of rapidly decreasing self-similar solutions, depending on a parameter $\beta$ with $0<\beta\le 1$. It is shown that if the initial data has compact support then the solution to the diffusive model at large time approximates the $\beta=1$ self-similar solution. This result supports the intuition that diffusion provides the mechanism whereby the $\beta=1$ self-similar solution of the LSW model is the only physically relevant one.