Global solution for the coagulation equation of water droplets in atmosphere between two horizontal planes

In this paper we give a global existence and uniqueness theorem for an initial and boundary value problem (IBVP) relative to the coagulation equation of water droplets and we show the convergence of the global solution to the stationary solution. The coagulation equation is an integro-differential equation that describes the variation of the density $\sigma$ of water droplets in the atmosphere. Furthermore, IBVP is considered on a strip limited by two horizontal planes and its boundary condition is such that rain fall from the strip. To obtain this result of global existence of the solution $\sigma$ in the space of bounded continuous functions, through the method of characteristics, we assume bounded continuous and small data, whereas the vector field, besides being bounded continuous, has $W^{1,\infty}-$ regularity in space.