| In this thesis we study the Jackson-Byrne's model(2000). The mathematical model describes the evolution of a vascular tumor in response to chemotherapeutic treatment. The model is based on mass conservation laws and on reaction-diffusion process within the tumor, Which is a free boundary problem for a system of partial differential equations governing intratumoral drug concentration and cancer cell density. The free boundary is the surface of the tumor. The unknown variables are the densities of the cells that are highly susceptible to the drug, the cells that have lower drug susceptibility, the intratumoral drug concentration, and the velocity of cells within the tumor as well as free boundary.Jackson and Byrne studied the model by approximation methods and numerical simulations. The purpose of this thesis is to establish a rigorous mathematical analysis of the model.In the first part of the thesis, we first transform the model into a problem in a fixed domain by a straightening transformation . Then , we prove the local existence and uniqueness of the solution of the system. Subsequently , we prove the global existence by extending the local solution to all t > 0. At last we explore a sufficient condition of the prescribed drug concentration in the tumor for successful treatment of the tumor.In the second part, we consider the stationary solution of the system. We first.claim the nonexistence of the stationary solution of the system. However, we can prove the existence of the quasi-stationary solution of the system. |
