Stationary solutions to coagulation-fragmentation equations

Abstract

Existence of stationary solutions to the coagulation-fragmentation equation is shown when the coagulation kernel $K$ and the overall fragmentation rate $a$ are given by $K(x, y) = x^αy^β+ x^βy^α$ and $a(x) = x^γ$, respectively, with $0\le α\le β\le1$, $α+β\in [0, 1)$, and $γ> 0$. The proof requires two steps: a dynamical approach is first used to construct stationary solutions under the additional assumption that the coagulation kernel and the overall fragmentation rate are bounded from below by a positive constant. The general case is then handled by a compactness argument.