Long-time behaviour of rouleau formation models

In this paper we study a two-component coagulation equation that models the aggregation of rouleaux in blood. We consider product kernels that have homogeneity $2$ and we characterize the initial data that lead to gelation. We prove that, when gelation occurs, the solution to the two-component coagulation equation localizes along a direction of the space of cluster as $t$ approaches the gelation time $0<T_{\ast }<\mathrm{\infty }$. The localization direction is determined by the initial datum. We also prove that the solution converges to a self-similar solution along the direction of localization.