Boundedness of the solution of a higher-dimensional parabolic-ODE-parabolic chemotaxis-haptotaxis model with generalized logistic source

This paper deals with a quasilinear chemotaxis-haptotaxis system with generalized logistic source {u(t) = del. (phi(u)del u) - del. (u del w) + u(1 - u(r) (1) - w), nu(t) = Delta nu - nu + u, (0.1) w(t) = -nu w, under homogeneous Neumann boundary conditions in a smooth bounded domain R-N(N >= 3), with parameter r > 1, where the given function phi(u) is the nonlinear diffusion. Besides appropriate smoothness assumptions, in this paper it is only required that phi(u) >= C-phi (u + 1)(m-1) for all u >= 0 with some C-phi > 0 and some m{> 2-2/N if 1 < r < N + 2/N, > 1+(N + 2 - 2r)(+)/ N + 2 if N + 2/2 >= N +2/N, >= 1 if r > N + 2/ 2 It is shown that then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a unique global classical solution that is uniformly bounded in ohm x (0, infinity).