A two-phase free boundary problem with discontinuous velocity: Application to tumor model

We consider a two-phase free boundary problem consisting of a hyperbolic equation for w and a parabolic equation for u, where w and u represent, respectively, densities of cells and cytokines in a simplified tumor growth model. The tumor region Ω(t) is enclosed by the free boundary Γ(t), and the exterior of the tumor, D(t), consists of a healthy normal tissue. Due to cancer cell proliferation, the convective velocity v→ of cells is discontinuous across the free boundary; the motion of the free boundary Γ(t) is determined by v→. We prove the existence and uniqueness of a solution to this system in the radially symmetric case for a small time interval 0≤t≤T, and apply the analysis to the full tumor growth model.