Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system

Abstract

In this paper, we consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-parabolic type \begin{equation*}\left\{\begin{split}&u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\qquad & x\in\Omega,\,\, t>0,\\&v_t=\Delta v+\alpha u-\beta v,\qquad &x\in\Omega, \,\,t>0,\\&w_t=\Delta w+\gamma u-\delta w,\qquad &x\in\Omega,\,\, t>0\end{split}\right.\end{equation*}under homogeneous Neumann boundary conditions, where $D(u)\geq c_D (u+\varepsilon)^{m-1}$ and $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary. It is shown that whenever $m>1$, for any sufficiently smooth nonnegative initial data, the system admits a global bounded classical solution for the case of non-degenerate diffusion (i.e., $\varepsilon>0$), while the system possesses a global bounded weak solution for the case of degenerate diffusion (i.e., $\varepsilon=0$).