Global boundedness of solutions to a chemotaxis–haptotaxis model with tissue remodeling

Abstract

We consider a cancer invasion model comprising a strongly coupled PDE–ODE system in two and three space dimensions. The system consists of a parabolic equation describing cancer cell migration arising from a combination of chemotaxis and haptotaxis, a parabolic/elliptic equation describing the dynamics of matrix degrading enzymes (MDEs), and an ODE describing the evolution and re-modeling of the extracellular matrix (ECM). We point out that this strongly coupled PDE–ODE setup presents new mathematical difficulties, which are overcome by developing new integral estimate techniques. We prove that the system admits a unique global classical solution which is uniformly bounded in time in the two-dimensional spatial setting at all cancer cell proliferation rates. We also prove that, in the case of three-dimensional convex spatial domain, when cancer cell proliferation is suitably small, the system also possesses a unique classical solution for appropriately small initial data. These results improve previously known ones.