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 (Escobedo et al. in Commun. Math. Phys. 231(1): 157–188, 2002; Escobedo et al. in J. Differ. Equ. 195(1): 143–174, 2003) and are counterparts to those for the continuous case studied in (Tran and Van in Commun. Pure Appl. Math. 75(6):1292–1331, 2022).