On a Keller–Segel–Stokes system with logistic type growth: blow-up prevention enforced by sublinear signal production

Abstract

In this paper, the coupled chemotaxis–fluid system $$\begin{aligned} \left\{ \begin{array}{lll} n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (n\nabla c)+rn-\mu n^2,\ \ \ &{}x\in \Omega ,\ t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+f(n),\ \ &{}x\in \Omega ,\ t>0,\\ u_t=\Delta u+\nabla P+n\nabla \phi ,\ \ &{}x\in \Omega ,\ t>0,\\ \nabla \cdot u=0,\ \ &{}x\in \Omega ,\ t>0\\ \end{array}\right. \end{aligned}$$ is considered associated with no-flux boundary conditions in a smooth bounded domain $$\Omega \subset \mathbb {R}^3$$ , where $$r\ge 0$$ and $$\mu >0$$ are given parameters. $$f\in C^1([0, \infty ))$$ is a given function satisfying $$f(s)\le Ks^{\alpha }$$ with $$K>0, \alpha >0$$ for all $$s>0$$ and $$f(0)\ge 0$$ . It is proved that whenever $$\begin{aligned} \frac{1}{2}<\alpha <1, \end{aligned}$$ for all $$r\ge 0$$ and $$\mu >0$$ , there exists at least a global classical solution that is uniformly bounded, which exhibits the mild saturation effect of signal production.