On Global Stability for Lifschitz-Slyozov-Wagner like equations

TL;DR

This paper proves weak global asymptotic stability for Lifschitz-Slyozov-Wagner like equations, extending previous local stability results and employing comparison with a quadratic model to analyze solutions with compact support.

Contribution

It establishes the first proof of weak global stability for LSW-like systems, broadening understanding of their long-term behavior.

Findings

Proves weak global asymptotic stability of LSW-like systems.

Uses comparison with a quadratic model for critical initial data.

Extends stability analysis beyond local results.

Abstract

This paper is concerned with the stability and asymptotic stability at large time of solutions to a system of equations, which includes the Lifschitz-Slyozov-Wagner (LSW) system in the case when the initial data has compact support. The main result of the paper is a proof of weak global asymptotic stability for LSW like systems. Previously strong local asymptotic stability results were obtained by Niethammer and Vel\'{a}zquez for the LSW system with initial data of compact support. Comparison to a quadratic model plays an important part in the proof of the main theorem when the initial data is critical. The quadratic model extends the linear model of Carr and Penrose, and has a time invariant solution which decays exponentially at the edge of its support in the same way as the infinitely differentiable self-similar solution of the LSW model.