Existence and uniqueness of the global solution for a mathematical model of the use of macrophages in tumor medicine

In this paper we study a free boundary problem modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumor sites. The model consists of five semi-linear parabolic partial differential equations. By applying the parabolic Schauder estimates and the Banach fixed point method, we prove that this system has a unique local classical solution. Using this result, parabolic LpLp estimates, the monotone method in the theory of reaction diffusion equations and the extension method, we further prove that if the tumor does not vanish at finite time then the solution exists for all t⩾0t⩾0.