Blow-up prevention by logistic source in an $ N $-D chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis

We consider the following fully parabolic Keller–Segel system with logistic source$ \left\{\begin{array}{ll} u_t = d_1\Delta u-\nabla\cdot(u\nabla v)+\nabla\cdot(u\nabla w)+u(a-b u^{m -1}), \\ {v_t = d_2\Delta v+\nabla\cdot(v\nabla w)- \mu v+\kappa u }, \quad\\ {w_t = d_3\Delta w+ru- \delta w}, \quad\ \end{array}\right. $over a bounded domain $ \Omega\subset\mathbb{R}^N(N\geq1) $, with smooth boundary $ \partial\Omega $, the parameters $ a\in \mathbb{R}, b, d_1, d_2, d_3, \mu, \kappa, r, \delta $ are all positive parameters. We proved that if $ m>1+\frac{N}{2} $, the corresponding Neumann initial-boundary value problem admits a unique global bounded classical nonnegative solution. One can also see that the blow up phenomenon can be prevented by the logistic source. To the best of our knowledge, there are the first results about the boundedness of solutions for the system with higher dimensional ($ N\geq3 $).