Tumor growth models with nutrient consumption

We study the saturation free boundary of the cell density and its interaction with the nutrient depletion interface in a coupled parabolic model for tumor growth with sublinear reactions. We establish a sharp quadratic--superquadratic growth dichotomy along the saturation interface and quantify the degeneration of quadratic profiles as they approach the joint saturation--depletion boundary. We further show that intrinsic regularity of the saturation interface can occur only at joint free boundary points, where half-space blow-ups of either component determine a unique joint blow-up profile. Finally, we prove a local backward coincidence between the joint and nutrient free boundaries near regular points, which yields spatial and temporal regularity of the joint interface.