Global existence to the discrete Safronov-Dubovski\v{i} coagulation equations and failure of mass-conservation

This paper presents the existence of global solutions to the discrete Safronov-Dubvoski\v{i} coagulation equations for a large class of coagulation kernels satisfying $\Lambda_{i,j} = \theta_i \theta_j + \kappa_{i,j}$ with $\kappa_{i,j} \leq A\theta_i \theta_j, \ \ \forall \ \ i,j\ge 1$ where the sequence $(\theta_i)_{i\geq 1}$ grows linearly or superlinearly with respect to $i$. Moreover, the failure of mass-conservation of the solution is also addressed which confirms the occurrence of the gelation phenomenon.