  In this paper, a reaction-diffusion system modeling injection of a
chemotherapeutic drug on the surface of a living tissue during a treatment for
cancer patients is studied. The system describes the interaction of the
chemotherapeutic drug and the normal, tumor and immune cells. We first
establish well-posedness for the nonlinear reaction-diffusion system, then
investigate the long-time behavior of solutions. Particularly, it is shown that
the cancer cells will be eliminated assuming that its reproduction rate is
sufficiently small in a short time period in each treatment interval. The
analysis is then essentially exploited to study an optimal drug injection rate
problem during a chemotherapeutic drug treatment for tumor cells, which is
formulated as an optimal boundary control problem with constraints. For this,
we show that the existence of an optimal drug injection rate through the
boundary, and derive the first-order optimality condition.
