Global existence to the discrete Safronov-Dubovskiǐ coagulation equations and failure of mass-conservation

This paper presents the existence of global solutions to the discrete Safronov-Dubvoskiǐ coagulation equations for a large class of coagulation kernels satisfying ${\mathrm{\Lambda }}_{i,j}={\theta }_i{\theta }_j+{\kappa }_{i,j}$ with ${\kappa }_{i,j}\le A{\theta }_i{\theta }_j,\mathrm{\forall }i,j\ge 1$ where the sequence $({\theta }_i{)}_{i\ge 1}$ 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.