Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion

We consider the chemotaxis- haptotaxis model {u(t) = del . (D(u)del u) - chi del .(u del v) - xi del . (u del w) + mu u(1 - u - w), x is an element of Omega, t > 0, v(t) = Delta v - v + u, x is an element of Omega, t > 0, w(t) = -vw, x is an element of Omega, t > 0 in a bounded smooth domain Omega subset of R-n (n >= 2), where chi, xi and mu are positive parameters, and the diffusivity D(u) is assumed to generalize the prototype D(u) = delta(u + 1)(-alpha) with alpha is an element of R. Under zero-flux boundary conditions, it is shown that for sufficiently smooth initial data (u(0), v(0), w(0)) and alpha < 2/n - 1, the corresponding initial-boundary problem possesses a unique global-in-time classical solution which is uniformly bounded. This paper develops some L-p-estimate techniques and thereby extends boundedness results in n <= 3 to arbitrary space dimensions. (C) 2015 Elsevier Inc. All rights reserved.