Existence, Regularity, and Stability in a Strongly Degenerate Nonlinear Diffusion and Haptotaxis Model of Cancer Invasion

We consider a mathematical model of cancer cell invasion of the extracellular matrix (ECM), comprising a strongly degenerate parabolic partial differential equation for the cell volume fraction, coupled with an ordinary differential equation for the ECM volume fraction (0 ≤ φ ≤ 1). The model captures the intricate link between the dynamics of invading cancer cells and the surround- ing ECM. First, migrating cells undergo haptotaxis, i.e., movement up ECM density gradients. Secondly, cancer cells degrade the ECM fibers by means of membrane-bound proteases. Finally, their migration speed is modulated by the ECM pore sizes, resulting in the saturation or even interruption of cell motility both at high and low ECM densities. The inclusion of the physical limits of cell migration results in two regimes of degeneracy (at φ = 0 and φ = 1) impacting simultaneously the nonlinear diffusion and haptotaxis terms.We devise specific estimates that are compatible with the strong degeneracy of the equation, focusing on the vacuum present at low ECM densities. Based on these regularity results, and associated local compactness, we prove stability of weak solutions with respect to perturbations on bounded initial data, and global existence of weak solutions in the limit of an approximate non-degenerate problem.