Global existence of classical solutions and numerical simulations of a cancer invasion model

In this paper, we study a cancer invasion model both theoretically and numerically. The model is a nonstationary, nonlinear system of three coupled partial differential equations modeling the motion of cancer cells, degradation of the extracellular matrix, and certain enzymes. We first establish existence of global classical solutions in both two- and three-dimensional bounded domains, despite the lack of diffusion of the matrix-degrading enzymes and corresponding regularizing effects in the analytical treatment. Next, we give a weak formulation and apply finite differences in time and a Galerkin finite element scheme for spatial discretization. The overall algorithm is based on a fixed-point iteration scheme. Our theory and numerical developments are accompanied by some simulations in two and three spatial dimensions.

Automated summary

In this paper, a cancer invasion model is studied both theoretically and numerically. The model consists of three coupled partial differential equations that describe the motion of cancer cells, degradation of the extracellular matrix, and certain enzymes. Global classical solutions are established in bounded domains of both two and three dimensions, despite the lack of diffusion of the matrix-degrading enzymes and corresponding regularizing effects. Finite difference and finite element methods are used for spatial discretization, and a fixed-point iteration scheme is employed for the overall algorithm. The theory and numerical developments are demonstrated through simulations in two and three dimensions.