Existence and Uniqueness of a Transport Equation coupled with a Balance law

In this article, we study the existence and uniqueness of a transport equation with discontinuous coefficients that comes from the solution of a balance law with a time-dependent source term. We also give a simplified proof providing an explicit formula for the solution of the balance law using a variational formulation. This system can be considered as a generalized version of the one-dimensional model for the large-scale structure of the formation of the universe.