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

Source abstract

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 $m\geq1$. Under the assumption that either $n\leq8$ and $m>(2n^2+4n-4)/(n^2+4n)$, or $n\geq9$ and $m>(2n^2+3n+2-\sqrt{8n(n+1)})/(n^2+2n)$ (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 $L^p$-estimate techniques that appear to be new in the context of chemotaxis-haptotaxis systems.