Generalized Moment Method for Smoluchowski Coagulation Equation and Mass Conservation Property

In this paper, we develop a generalized moment method with continuous weight function for the Smoluchowski coagulation equation in its form to study mass conservation property of equation. We first establish some basic inequalities and prove under sufficient condition on kernel an initial condition, utilizing these inequalities. Additionally, provide concrete examples kernels that exhibit properties show either polynomial or exponential growth along specific particular curves.