Solutions with peaks for a coagulation-fragmentation equation. Part I: stability of the tails

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behavior is stable. In a companion paper, we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.