Coagulation equations with source leading to anomalous self-similarity

Abstract

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 γ ⩽ − 1 , and behaves like x γ + λ y − λ when y ≪ x with γ + 2 λ > 1 . Our analysis shows that the long-time behaviour of the solutions depends on the parameters γ and λ. 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 γ + λ and on whether γ = − 1 or γ < − 1 . In all these cases, the scaling differs from the usual one that has been previously obtained when γ + 2 λ < 1 or γ + 2 λ ⩾ 1 , γ > − 1 . In the last part of the paper, we present some conjectures supporting the self-similar ansatz also for the critical case γ + 2 λ = 1 , γ ⩽ − 1 .