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In this paper, we consider a model with tumor microenvironment involvingnutrient density, extracellular matrix and matrix-degrading enzymes, whichsatisfy a coupled system of PDEs with a free boundary. For this coupledparabolic-hyperbolic free boundary problem, we prove that there is a uniqueradially symmetric solution globally in time. The stationary problem involvesan ODE system which is transformed into a singular integro-differentialequation. We establish a well-posed theorem for such general types of equationsby the shooting method; the theorem is then applied to our problem for theexistence of a stationary solution. In addition, for this highly nonlinearproblem, we also prove the uniqueness of the stationary solution, which is anontrivial result. In addition, numerical simulations indicate that thestationary solution is likely locally asymptotically stable for reasonablerange of parameters.

A parabolic-hyperbolic system modeling the growth of a tumor