Coagulation equations with source leading to anomalous self-similarity

Abstract

We study the long-time behaviour of the solutions to Smoluchowski coagulation equations with a source term of small clusters. The source drives the system out-of-equilibrium, leading to a rich range of different possible long-time behaviours, including anomalous self-similarity. The coagulation kernel is non-gelling, homogeneous, with homogeneity $\gamma \leq -1 $, and behaves like $x^{\gamma+\lambda} y^{-\lambda} $ when $y \ll x$ with $\gamma+2\lambda > 1 $. Our analysis shows that the long-time behaviour of the solutions depends on the parameters $\gamma$ and $\lambda$. More precisely, we argue that the long-time behaviour is self-similar, although the scaling of the self-similar solutions depends on the sign of $\gamma+\lambda$ and on whether $\gamma=-1$ or $\gamma < -1$. In all these cases, the scaling differs from the usual one that has been previously obtained when $\gamma+2\lambda <1$ or $\gamma+2\lambda \geq 1, \gamma >-1$. In the last part of the paper, we present some conjectures supporting the self-similar ansatz also for the critical case $\gamma+2\lambda=1, \gamma \leq -1 $.