Article Contents

2018, Volume 11, Issue 4: 933-952. Doi: 10.3934/krm.2018037

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Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel

1.
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, 118 route de Narbonne, F-31062 Toulouse cedex 9, France
2.
Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Received: May 2017

Revised: December 2017

Published: April 2018

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Abstract

We characterize the long-time behaviour of solutions to Smoluchowski's coagulation equation with a diagonal kernel of homogeneity $γ < 1$. Due to the property of the diagonal kernel, the value of a solution at a given cluster size 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.

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Mathematics Subject Classification: Primary: 70F99, 82C22, 45M05.

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