A parabolic-hyperbolic system modeling the growth of a tumor

TL;DR

This paper develops and analyzes a coupled parabolic-hyperbolic PDE model with a free boundary to describe tumor growth, proving existence, uniqueness, and stability of solutions, supported by numerical simulations.

Contribution

It introduces a novel tumor growth model involving a free boundary and proves key mathematical properties including existence, uniqueness, and stability of solutions.

Findings

Unique global radially symmetric solutions are established.

Existence and uniqueness of stationary solutions are proved.

Numerical simulations suggest local stability of stationary solutions.

Abstract

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.