# American Institute of Mathematical Sciences

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February  2021, 41(2): 747-775. doi: 10.3934/dcds.2020299

## Large time behavior of exchange-driven growth

 1 WMG, University of Warwick, UK, Mathematics Institute, University of Warwick, UK 2 Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60 (room 2.023), D-53115 Bonn, Germany

*Corresponding author: E.esenturk.1@warwick.ac.uk

Received  April 2019 Revised  April 2020 Published  August 2020

Exchange-driven growth (EDG) is a model in which pairs of clusters interact by exchanging single unit with a rate given by a kernel $K(j, k)$. Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel $(i)$ $K(j, k) = b_{j}a_{k}$ and $(ii)$ $K(j, k) = ja_{k}+b_{j}+\varepsilon\beta_{j}\alpha_{k}$. For type I kernels, under the detailed balance assumption, we show that the system admits unique equilibrium up to a critical mass $\rho_{s}$ above which there is no equilibrium. We prove that if the system has an initial mass below $\rho_{s}$ then the solutions converge to a unique equilibrium distribution strongly where if the initial mass is above $\rho_{s}$ then the solutions converge to cricital equilibrium distribution in a weak sense. For type II kernels, we do not make any assumption of detailed balance and equilibrium is shown as a consequence of contraction properties of solutions. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we prove exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm.

Citation: Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299
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