关键词: boundedness, convection, chemotaxis, tumor invasion

Abstract: This paper is concerned with the parabolic-parabolic-elliptic system $\begin{equation*} \begin{cases} u_t=\Delta u-\chi \nabla \cdot \left(u\nabla v\right) +\xi_1\nabla \cdot \left(u^m\nabla w\right), &x\in \Omega,t>0,\\ v_t=\Delta v+\xi_2 \nabla \cdot \left(v\nabla w\right)+u-v,&x\in \Omega,t>0,\\[2mm] 0=\Delta w+u-\frac{1}{|\Omega|}\int_\Omega u, \int_\Omega w=0,&x\in \Omega,t>0,\\[3mm] \frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=\frac{\partial w}{\partial\nu}=0, &x\in\partial\Omega, t>0,\\[2mm] u(x,0)=u_0(x),v(x,0)=v_0(x), &x\in \Omega \end{cases} \end{equation*}$ in a bounded domain $\Omega \subset \mathbb{R}^{n}$ with a smooth boundary, where the parameters $\chi,\xi_1,\xi_2 $ are positive constants and $m\geq 1$. Based on the coupled energy estimates, the boundedness of the global classical solution is established in any dimensions ($n\geq 1$) provided that $m>1$.

Key words: boundedness, convection, chemotaxis, tumor invasion