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=\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\\ \endaligned\right. \end{align*} on a bounded domain $\Omega\subset \mathbb{R}^{N}$ ($N\geq1$) 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) $N\geq4$ 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) $N=2$, $\alpha\geq2$, 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.