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 $K(x,y)=x^{\alpha }{y}^{\beta }+x^{\beta }{y}^{\alpha }$ and $a(x)=x^{\gamma }$, respectively, with $0\le \alpha \le \beta \le 1$, $\alpha +\beta \in [0,1)$, and $\gamma >0$. 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.