Free Boundary Problems for Tumor Growth: a Viscosity Solutions Approach

The mathematical modeling of tumor growth leads to singular “stiff pressure law” limits for porous medium equations with a source term. Such asymptotic problems give rise to free boundaries, which, in the absence of active motion, are generalized Hele-Shaw flows. In this note we use viscosity solutions methods to study limits for porous medium-type equations with active motion. We prove the uniform convergence of the density under fairly general assumptions on the initial data, thus improving existing results. We also obtain some additional information/regularity about the propagating interfaces, which, in view of the discontinuities, can nucleate and, thus, change topological type. The main tool is the construction of local, smooth, radial solutions which serve as barriers for the existence and uniqueness results as well as to quantify the speed of propagation of the free boundary propagation. Key-words: Elliptic-Parabolic problems; viscosity solutions; free boundary; Tumor growth; AMS Class. No: 35K55; 35B25; 35D40; 76D27;