Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type

Abstract

This paper deals with the global existence and boundedness of solutions to the following quasilinear attraction-repulsion chemotaxis system: $$ \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u_{t}=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u\nabla v)+\nabla\cdot (\xi u\nabla w), & x\in\Omega, 0=\Delta v+\alpha u-\beta v,& x\in\Omega, 0=\Delta w+\gamma u-\delta w,& x\in\Omega, t>0, \end{array}\displaystyle \right . $$ under homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^{n}$ ( $n\geq2$ ) with smooth boundary, where $D(u)\geq c_{D} u^{m-1}$ with $m\geq1$ and some constant $c_{D}>0$ . It is proved that if $\xi\gamma-\chi\alpha>0$ or $m>2-\frac{2}{n}$ , then for any sufficiently regular initial data, this system possesses a unique global bounded classical solution for the case of nondegenerate diffusion (i.e., $D(u)>0$ for all $u\geq 0$ ), whereas for the case of degenerate diffusion (i.e., $D(u)\geq0$ for all $u\geq0$ ), it is shown that there exists a global bounded weak solution under the same assumptions.