A free boundary analysis of tumor invasion driven by angiogenesis

We discuss a free boundary model for tumor invasion that describes a cloud of cells that diffuse and, at the same time, are drifted along the vector field of the chemotactic direction. The model captures the evolution of a solid tumor, including the process of angiogenesis, which consists in the formation of new blood vessels that supply the tumor with oxygen and other nutrients, thereby promoting its spread and growth.
We prove that, once formed, the tumor survives through time, maintaining strictly positive thickness. An explicit expression in terms of the initial data is derived.
Moreover, we distinguish two regimes depending on the ratio $\kappa$ between the spreading of tumor cells and the growth of the tumor mass. If $\kappa$ is sufficiently large, then the tumor grows exponentially in time and invades the entire host tissue. In contrast, if $\kappa$ is small enough, then either the tumor remains bounded in size over time or may experience a fast contraction.