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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 and , respectively, with , \alpha + \beta \in [0,1\right., and . 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.
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Philippe Laurençot, Stationary solutions to coagulation-fragmentation equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 7, pp. 1903–1939DOI 10.1016/J.ANIHPC.2019.06.003