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

This paper deals with a boundary-value problem for a coupled quasilinear chemotaxis--haptotaxis model with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi \nabla\cdot(u\nabla w)+\mu u(1-u-w),\\ v_t=\Delta v- v +u,\quad \\ w_t=- vw\\ \end{array}\right.$$ in $N$-dimensional smoothly bounded domains, where the parameters $\xi ,\chi> 0$, $\mu> 0$. The diffusivity $D(u)$ 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 is proved 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 proper regularity hypotheses on the initial data, the corresponding initial-boundary problem 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 of possibly degenerate diffusion), the existence of bounded weak solutions for system is shown. This extends some recent results by several authors.