A Chemotaxis-Haptotaxis Model: The Roles of Nonlinear Diffusion and Logistic Source

This paper deals with a chemotaxis-haptotaxis model of cancer invasion of tissue (extracellular matrix (ECM) ). The model consists of a parabolic chemotaxis-haptotaxis PDE describing the evolution of cancer cell density, a parabolic PDE governing the evolution of matrix degrading enzyme concentration, and an ODE reflecting the degradation of ECM. Following a recent approach proposed by Szymańska, Morales-Rodrigo, Lachowicz, and Chaplain Math. Models Methods Appl. Sci. , 19 (2009), pp. 257–281, we assume that the migration of cancer cells through ECM is more like movement in a porous medium. Accordingly, we consider the self-diffusion coefficient D (u) of cancer cells to be a nonlinear function generalizing the prototype D (u) = (u+1) ^m-1 for some m1. Under the assumption that either n8 and m> (2n²+4n-4) / (n²+4n), or n9 and m> (2n²+3n+2-8n (n+1) ) / (n²+2n) (where n denotes the space dimension), and in the presence of logistic dampening of cancer cell densities, the global existence of a unique classical solution to the model is proved by developing some Lᵖ-estimate techniques that appear to be new in the context of chemotaxis-haptotaxis systems.