Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel

TL;DR

This paper studies the long-term behavior of solutions to Smoluchowski's coagulation equation with a diagonal kernel, revealing that solutions generally oscillate periodically between rescaled self-similar forms, with specific conditions for convergence.

Contribution

It characterizes the long-time dynamics of solutions with diagonal kernels, showing periodic oscillations and providing conditions for convergence to self-similarity.

Findings

Solutions exhibit periodic oscillations over time.

Convergence to self-similar solutions depends on initial data.

Uniqueness of self-similar profiles is established.

Abstract

We characterize the long-time behaviour of solutions to Smoluchowski's coagulation equation with a diagonal kernel of homogeneity $\gamma < 1$. Due to the property of the diagonal kernel, the value of a solution depends only on a discrete set of points. As a consequence, the long-time behaviour of solutions is in general periodic, oscillating between different rescaled versions of a self-similar solution. Immediate consequences of our result are a characterization of the set of data for which the solution converges to self-similar form and a uniqueness result for self-similar profiles.