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This paper deals with a boundary-value problem for a coupled quasilinearchemotaxis--haptotaxis model with nonlinear diffusion$$\left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nablau)-\chi\nabla\cdot(u\nabla v)-\xi \nabla\cdot(u\nabla w)+\mu u(1-u-w),\\v_t=\Delta v- v +u,\quad \\ w_t=- vw\\ \end{array}\right. $$ in $N$-dimensionalsmoothly bounded domains, where the parameters $\xi ,\chi> 0$, $\mu> 0$. Thediffusivity $D(u)$ is assumed to satisfy $D(u)\geq C_{D}u^{m-1}$ for all $u >0$ with some $C_D>0$. Relying on a new energy inequality, in this paper, it isproved that under the conditions$$m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_0\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_0\|_{L^\infty(\Omega)})-\mu)_{+}}+1)(N+\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_0\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_0\|_{L^\infty(\Omega)})-\mu)_{+}}-1)}{N,$$ and properregularity hypotheses on the initial data, the corresponding initial-boundaryproblem possesses at least one global bounded classical solution when $D(0) >0$ (the case of non-degenerate diffusion), while if, $D(0)\geq 0$ (the case ofpossibly degenerate diffusion), the existence of bounded weak solutions forsystem is shown. This extends some recent results by several authors.

A new result for boundedness of solutions to a quasilinear higher-dimensional chemotaxis -- haptotaxis model with nonlinear diffusion