Wellposedness of solution for an $N$-D chemotaxis-convection model during tumor angiogenesis

Abstract

In this paper, we consider the following parabolic-parabolic-elliptic system } \begin{align*} \left\{\aligned & u_t=Δu-\nabla\cdot(u\nabla v)+ξ\nabla\cdot(u\nabla w)+au-μu^α, && x\inΩ, t>0,\\ & v_t=Δv+\nabla\cdot(v\nabla w)-v+u,&& x\inΩ, t>0,\\ & 0=Δw-w+u,&& x\inΩ, t>0\\ \endaligned\right. \end{align*} on a bounded domain $\Omega \subset {\mathbb{R}}^N$ ($N\ge 1$) with smooth boundary $\partial \Omega$, where $\mu$, $a$, $\alpha$ are positive constants and $\xi \in \mathbb{R}$. 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.