Well-posedness of a model equation for neurotransmitter diffusion with reactive boundaries

We consider a diffusion equation with reactive boundary conditions. The equation is a model equation for the diffusion of classical neurotransmitters in the tortuous space between cells in the brain. The equation determines the concentration of neurotransmitters such as glutamate and GABA (gamma-aminobutyrate) and the probability for neurotransmitter molecules to be immobilized by binding to protein molecules (receptors and transporters) at the cell boundary (cell membrane). On a regularized problem, we derive a priori estimates. Then, by a compactness argument, we show the existence of solutions. By exploiting the particular structure of the boundary reaction terms, we are able to prove that the solutions are unique and continuous with respect to initial data.