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

Abstract

This paper deals with a boundary-value problem for a coupled quasilinear chemotaxis--haptotaxis model with nonlinear diffusion $$\{\begin{array}{l}{u}_t=\nabla \cdot (D(u)\nabla u)-\chi \nabla \cdot (u\nabla v)-\xi \nabla \cdot (u\nabla w)+\mu u(1-u-w),\\ v_t=\Delta v-v+u,\\ w_t=-vw\end{array}$$ in $N$-dimensional smoothly bounded domains, where the parameters $\xi ,\chi >0$, $\mu >0$. The diffusivity $D(u)$ is assumed to satisfy $D(u)\ge C_D{u}^{m-1}$ 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 $$m>\frac{2N}{N+\frac{(\frac{{\max }_{s\ge 1}{\lambda }_0^{\frac{1}{s+1}}(\chi +\xi \Vert w_0{\Vert }_{L^{\infty }(\Omega )})}{({\max }_{s\ge 1}{\lambda }_0^{\frac{1}{s+1}}(\chi +\xi \Vert w_0{\Vert }_{L^{\infty }(\Omega )})-\mu {)}_{+}}+1)(N+\frac{{\max }_{s\ge 1}{\lambda }_0^{\frac{1}{s+1}}(\chi +\xi \Vert w_0{\Vert }_{L^{\infty }(\Omega )})}{({\max }_{s\ge 1}{\lambda }_0^{\frac{1}{s+1}}(\chi +\xi \Vert w_0{\Vert }_{L^{\infty }(\Omega )})-\mu {)}_{+}}-1)}{N}},$$ 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, $D(0)\ge 0$ (the case of possibly degenerate diffusion), the existence of bounded weak solutions for system is shown. This extends some recent results by several authors.

References in corpus (1)

• Boundedness in a two-dimensional chemotaxis-haptotaxis system