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We study a critical case of Coagulation-Fragmentation equations withmultiplicative coagulation kernel and constant fragmentation kernel. Our methodis based on the study of viscosity solutions to a new singular Hamilton-Jacobiequation, which results from applying the Bernstein transform to the originalCoagulation-Fragmentation equation. Our results include wellposedness,regularity and long-time behaviors of viscosity solutions to theHamilton-Jacobi equation in certain regimes, which have implications towellposedness and long-time behaviors of \emph{mass-conserving} solutions tothe Coagulation-Fragmentation equation.

Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel