Global existence and boundedness to an N-D chemotaxis-convection model during tumor angiogenesis

In this paper, we consider the following parabolic–parabolic–elliptic system $\left(\begin{array}{cccc}& u_t=\Delta u-\nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w)+au-\mu u^{\alpha },& & x\in \Omega ,t>0,\\ & v_t=\Delta v+\nabla \cdot (v\nabla w)-v+u,& & x\in \Omega ,t>0,\\ & 0=\Delta w-w+u,& & x\in \Omega ,t>0\\ \end{array}\right)$on a bounded domain $\Omega \subset R^N$ ($N\ge 1$) with smooth boundary $\partial \Omega$, where $\mu$, $a$, $\alpha$ are positive constants and $\xi \in R$. If one of the following cases holds:

(i) $N\ge 4$ and $\alpha >\frac{4N-4+N\sqrt{2N^2-6N+8}}{2N}$;

(ii) $N=3$, $\alpha >2$, for any $\mu >0$ or $\alpha =2$, the index $\mu$ should be suitably big;

(iii) $N=2$, $\alpha \ge 2$, for any $\mu >0$.

Without any restriction on the index $\xi$, for any given suitably regular initial data, the corresponding Neumann initial–boundary problem admits a unique global and bounded classical solution.