The process of growth related to coagulation is described by the following mathematical equations: \begin{equation} \partial_{t}c(t,x) + \partial_{x}(gc)(t,x) = Q(c)(t,x) \quad \text{for } (t,x) \in (0,\infty)^2, \qquad c(0,x) = c_0(x) \geq 0, \end{equation} where \begin{equation} Q(c)(t,x) = \frac{1}{2} \int_{0}^{x} K(x-y,y)c(t,x-y)c(t,y)dy - \int_{0}^{\infty} K(x,y)c(t,x)c(t,y)dy. \end{equation} These mathematical formulations find applications across various domains, such as phytoplankton aggregation, polymer formation, industrial powder production, cloud droplet formation, and the formation of celestial bodies in astrophysics.

This presentation addresses the existence and uniqueness of solutions to the growth–coagulation equation, with particular emphasis on coagulation kernels that are singular near the origin and exhibit at most linear growth at infinity. The existence of weak solutions is established using the method of characteristics together with an $L^{1}$-weak compactness argument, relying on the Banach fixed-point theorem and a refined version of the Arzela–Ascoli theorem. Furthermore, we prove the continuous dependence of solutions on the initial data by employing the DiPerna–Lions theory, Gronwall’s inequality, and suitable moment estimates. As a consequence, uniqueness of solutions is obtained. These results significantly extend several earlier contributions in the literature by allowing more general coagulation kernels with singularities.