A mathematical model of the response of spatially structured tumors to chemotherapy: drug kinetics is studied. The model is a free boundary problem of a partial differential equation. The tumor is assumed to comprise a single cell population which reproduces and dies at a rate dependent on the local drug concentration. By using the method of the L^ptheory for parabolic equations, the Banach fixed point theorem and the extensions method under some general conditions, it is proved that this problem has a unique global solution. And then, by applying upper and lower solution method in the theory of reaction diffusion equations under some other general conditions to obtain the stationary solution. It is proved that if 0<w≤w*,there is no staionary solution;If w*<w<w¯^,there is a unique staionary solution(w_s(r) ,R_s)(R_s>0).