In this notes we consider a mathematical model of parabolic type which describes the pollutant on the surface of a narrow channel, along with a water stream. The polluted flows occurs very often in different technological processes and in the nature, especially when the flow is an aggressive liquid. This is a parabolic equation with initial and boundary condition (IBVP) which includes a reaction nonlinearity of functional type. We discuss some methods for existence of classical solution, blow-up and weak solutions.