Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion

This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion {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, under homogeneous Neumann boundary conditions in a bounded smooth domain Omega subset of R-n, n = 2,3, 4, where chi, xi and mu are given nonnegative parameters. The diffusivity D(u) is assumed to satisfy D(u) >= delta um(-1) for all u > 0 with some delta > 0. It is proved that for sufficiently regular initial data global bounded solutions exist whenever m> 2- 2/n. For the case of non- degenerate diffusion (i.e. D(0) > 0) the solutions are classical; for the case of possibly degenerate diffusion (D(0) >= 0), the existence of bounded weak solutions is shown.