The spatially inhomogeneous coagulation-condensation process is an interesting topic of study as the phenomenon’s mathematical aspects mostly undiscovered and has multitudinous empirical applications. In this present exposition, we exhibit the existence of a continuous solution for the corresponding model with the following \emph{singular} type coagulation kernel: \[K(x,y)~\le~\frac{\left( x + y\right)^\theta}{\left(xy\right)^\mu}, ~~\text{for} ~x, y \in (0,\infty), \text{where}~ \mu \in \left[0,\tfrac{1}{2}\right] \text{ and } ~\theta \in [0, 1].\] The above-mentioned form of the coagulation kernel includes several practical-oriented kernels. Finally, uniqueness of the solution is also investigated.

This article is devoted to the study of existence of a mass conserving global solution for the collision-induced nonlinear fragmentation model which arises in particulate processes, with the following type of collision kernel: \[C(x,y)~\le~k_1 \frac{(1 + x)^\nu (1 + y)^\nu}{\left(xy\right)^\sigma},\] for all ~$x, y \in (0,\infty)$, where $k_1$ is a positive constant, $\sigma \in \left[0,\tfrac{1}{2}\right]$ and $\nu \in [0, 1]$. The above-mentioned form includes many practical oriented kernels of both \emph{singular} and \emph{non-singular} types. The singularity of the unbounded collision kernel at coordinate axes extends the previous existence result of Paul and Kumar [Mathematical Methods in the Applied Sciences 41 (7) (2018) 2715–2732 (\href{https://doi.org/10.1002/mma.4775}{doi:10.1002/mma.4775})] and also exhibits at most quadratic growth at infinity. Finally, uniqueness of solution is also investigated for pure singular collision rate, i.e., for ~ $\nu=0$.

This article provides mathematical proof of the existence of stationary solutions for the coagulation equation including source and efflux terms. We demonstrate the convergence of time dependent solutions to these stationary solutions and highlight the exponential rate of convergence. These properties are analyzed for affine linear coagulation kernels, non-negative source terms and positive efflux rates. Numerical examples are included to demonstrate the predicted convergence behaviour.