Global boundedness and large time behavior of solutions to a chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis

In this paper, we investigate a parabolic–parabolic–elliptic system that describes the initial stage of tumor-related angiogenesis, given by \(\begin{aligned} \left\{ \begin{array}{ll} u_t=\Delta u-\nabla \cdot (u\nabla v)+\xi \nabla \cdot (u^m\nabla w)+\mu u(1-u^\alpha ),\\ v_t=\Delta v+\chi \nabla \cdot (v\nabla w)-v+u,\\ 0=\Delta w-w+u. \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t=\mathrm{\Delta }u-\mathrm{\nabla }·(u\mathrm{\nabla }v)+\xi \mathrm{\nabla }·(u^m\mathrm{\nabla }w)+\mu u(1-u^{\alpha }),\hfill \\ v_t=\mathrm{\Delta }v+\chi \mathrm{\nabla }·(v\mathrm{\nabla }w)-v+u,\hfill \\ 0=\mathrm{\Delta }w-w+u.\hfill \end{array}\right)\end{array}$$ We demonstrate that the model possesses a global classical solutions for all suitably regular initial data and associated homogeneous Neumann boundary conditions. Additionally, when m=1, the asymptotic behavior can be investigated.