Boundedness in a quasilinear chemotaxis-haptotaxis model of parabolic-parabolic-ODE type

This paper deals with the boundedness of solutions to the following quasilinear chemotaxis-haptotaxis model of parabolic-parabolic-ODE type: {u(t) = del . (D(u)del u) - chi del . (u del v) - xi del . (u del w) + mu u(1 - u(r-1) - w), x is an element of Omega, t > 0, v(t) = Delta v - v + u(eta), x is an element of Omega, t > 0, w(t) = -vw, x is an element of Omega, t > 0, under zero-flux boundary conditions in a smooth bounded domain Omega subset of R-n(n >= 2), with parameters r >= 2, eta is an element of (0, 1] and the parameters chi > 0, xi > 0, mu > 0. The diffusivity D(u) is assumed to satisfy D(u) >= delta u(-alpha), D(0) > 0 for all u > 0 with some alpha is an element of R and delta > 0. It is proved that if alpha < n+2-2n eta/2+n, then, for sufficiently smooth initial data (u(0), v(0), w(0)), the corresponding initial-boundary problem possesses a unique global-in-time classical solution which is uniformly bounded.