We consider a procedure for cancer therapy which consists of injecting replication-competent viruses into the tumor. The viruses infect tumor cells, replicate inside them, and eventually cause their death. As infected cells die, the viruses inside them are released and then proceed to infect adjacent tumor cells. However, a major factor influencing the efficacy of virus agents is the immune response that may limit the replication and spread of the replication-competent virus. The competition between tumor cells, a replication-competent virus and an immune response is modelled as a free boundary problem for a nonlinear system of partial differential equations, where the free boundary is the surface of the tumor. In this model, the immune response equation is a semilinear parabolic equation, including a chemotaxis term which is used to describe the movement of the immune response induced by gradients of the infected cell density. Under the assumption that the chemotactic sensitivity coefficient is small compared with the diffusion coefficient of the immune response, we prove the global existence and uniqueness of the solution of this free boundary problem. For large chemotactic coefficient, the global existence is still open.
Global existence of solutions of the free boundary problem for the equations of magnetohydrodynamic compressible fluid