Stationary solutions to coagulation-fragmentation equations

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$0\le \alpha \le \beta \le1$,$\alpha+\beta\in [0, 1)$,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.