Existence of solutions to the continuous RednerBen-AvrahamKahng coagulation equation

We take into account a coagulation model that simulates a distinct kind of dynamics. In this model, two particles collide to produce a single particle, but the resulting particle decreases in size, allowing each particle to be fully identified by its size. It is demonstrated that the corresponding evolving integral partial differential equation has solutions for product-type coagulation kernels i.e. $0 \le \mathfrak{K}(\varrho, \varsigma)= \mathfrak{K}(\varsigma, \varrho)=r(\varsigma) r(\varrho)+\alpha(\varsigma, \varrho), (\varsigma, \varrho)\in \mathbb{R}_+^2$ and $\sup_{\varsigma \in [0,R]} \frac{\mathfrak{K}(\varsigma, \varrho)}{\varrho} \to 0 \ \mbox{as} \ \varrho \to \infty$.