Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source

In this paper, we consider the quasilinear chemotaxis-haptotaxis system {u(t) = del . (D(u)del u) - del . (S-1(u)del v) - del . (S-2(u)del w) + uf(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 >= 1) under zero-flux boundary conditions, where the nonlinearities D, S1 and S2 are assumed to generalize the prototypes D(u) = CD(u + 1)(m-1), S-1(u) = C(S1)u(u + 1)(q1-1) and S-2(u) = C(S2)u(u + 1)(q2-1) with C-D, C-S1, C-S2 > 0, m, q(1), q(2) is an element of R and f(u, w) is an element of C-1([0,+ infinity) x [0,+ infinity)) fulfills f(u, w) <= r - bu for all u >= 0 and w >= 0, where r > 0, b> 0. Assuming nonnegative initial data u(0)(x) is an element of W-1,W-infinity(Omega), v(0)(x) is an element of W-1,W-infinity(Omega) and w(0)(x) is an element of C-2,C-alpha(Omega) O) for some a. (0, 1), we prove that (i) for n <= 2, if max{q(1), q(2)} < m+ 2/n - 1, then (star) has a unique nonnegative classical solution which is globally bounded, (ii) for n > 2, if max{q(1), q(2)} < m+ 2/n - 1 and m > 2 - 2/n or max{q(1), q(2)} < m + 2/n - 1 and m <= 1, then (star) has a unique nonnegative classical solution which is globally bounded.