SOME FURTHER PROGRESS FOR BOUNDEDNESS OF SOLUTIONS TO A QUASILINEAR HIGHER-DIMENSIONAL CHEMOTAXIS-HAPTOTAXIS MODEL WITH NONLINEAR DIFFUSION

This study focuses on the N -dimensional chemotaxis-haptotaxis model with nonlinear diffusion that was initially proposed by Chaplain and Lolas (see [9]) to describe the interactions between cancer cells, the matrixdegrading enzyme, and the host tissue during cancer cell invasion. Accordingly, we consider the diffusion coefficient D(u) of cancer cells to be a nonlinear function satisfying D(u) >= C(D)u(m-1)for all u > 0 with some C-D > 0 and m > 0. Relying on a new energy inequality and iteration argument, this paper proves that under the mild condition m > 2N (N + 1)[max 2 <= s <= N+2 +2 lambda (1s)(0),(s)(chi + xi parallel to omega(0)parallel to L infinity((Omega)))-mu],/ (N + 2)[(N + 1) max2 <= s <= N+2 lambda (1s) (0,s) (chi + xi parallel to omega(0)parallel to L infinity((Omega)))- N mu] +; and proper regularity hypotheses on the initial data, the corresponding initialboundary problem has at least one globally bounded classical solution when D (0) > 0 (the case of nondegenerate diffusion), while if D(0) >= 0 (the case of possibly degenerate diffusion), the existence of bounded weak solutions for the system is shown, where the positive parameters xi, chi, and mu > 0 measure the chemotactic and haptotactic sensitivities and proliferation rate of the cells, respectively.