  In this paper, we consider a model with tumor microenvironment involving
nutrient density, extracellular matrix and matrix-degrading enzymes, which
satisfy a coupled system of PDEs with a free boundary. For this coupled
parabolic-hyperbolic free boundary problem, we prove that there is a unique
radially symmetric solution globally in time. The stationary problem involves
an ODE system which is transformed into a singular integro-differential
equation. We establish a well-posed theorem for such general types of equations
by the shooting method; the theorem is then applied to our problem for the
existence of a stationary solution. In addition, for this highly nonlinear
problem, we also prove the uniqueness of the stationary solution, which is a
nontrivial result. In addition, numerical simulations indicate that the
stationary solution is likely locally asymptotically stable for reasonable
range of parameters.
