Theory of Evolution Systems Applied to a Cosmic Ray Diffusion Model

In this article we study a model constituted by three integro-differential equations describing the densities of nucleons and of two groups of mesons (depending on their energy—high or low) diffusing in the Earth’s atmosphere. We assume that the secondary particles, produced after the collisions in air of the primary ones with particles in the air, have a lower energy with respect to the incident ones; indeed the transformation of mass into energy is considered not to be relevant by astrophysicists. We list assumptions that allow us to prove existence and uniqueness of non-negative solutions for the densities through generalization of the theory of evolution operators and evolution systems in Banach spaces. In the special case in which certain operators are bounded, the densities can be written as series of the form ∑ i xiFi (E) where the factors Fi (E) are given by the actions of suitable linear operators on the initial datum, and it is also proved that these can be obtained by recurrent formulas. The truncation of the series allows the estimate of the corresponding committed error.