Boundedness for the chemotaxis system with logistic growth

In this paper, we consider a mathematical model motivated by the studies of coral broadcast spawning$\{\begin{array}{cc}{\partial }_{t}n+u\cdot \nabla n-\Delta n& =-\chi \nabla \cdot \left(n\nabla c\right)+n-ϵn^q\\ {\partial }_{t}c+u\cdot \nabla c-\Delta c& =-c+n\end{array}\text{ in }{R}^d\times R^{+},$ where $d=2,3$, $ϵ>0$, and $q\ge 2$. We establish global-in-time well-posedness and boundedness of the solution to the Cauchy problem of this system by developing local-in-space estimates. The crux point of our proof depends intensely on localization in the space of solutions induced by “local effect” of the $L^{\infty }\left(R^d\right)$-norm.