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

Here, we study a discrete Coagulation-Fragmentation equation with a multiplicative coagulation kernel and a constant fragmentation kernel, which is critical. We apply the discrete Bernstein transform to the original Coagulation-Fragmentation equation to get two new singular Hamilton-Jacobi equations and use viscosity solution methods to analyze them. We obtain well-posedness, regularity, and long-time behaviors of the viscosity solutions to the Hamilton-Jacobi equations in certain ranges, which imply the well-posedness and long-time behaviors of mass-conserving solutions to the Coagulation-Fragmentation equation. The results obtained provide some definitive answers to a conjecture posed in [11,10], and are counterparts to those for the continuous case studied in [32].