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.