  In this paper we consider an optimal control problem arising from a
chemotherapeutic drug treatment for tumor cells in a living tissue. The
mathematical model for the interaction of chemotherapeutic drug and the normal,
tumor and immune cells are governed by a nonlinear reaction-diffusion system.
We first establish the well-posedness for the nonlinear system. Then we study
the long-time behavior of the solution for the model problem. Finally, we
design an optimal drug dosage, which leads to an optimal control problem under
certain constrains. A complicated factor for the optimal control problem is
that a minimum level of normal cells in patients must be maintained during a
treatment. It is shown that there is an optimal drug dosage for the
chemotherapeutic treatment for patients.
