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

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.