Approaching constant steady states in a Keller-Segel-Stokes system with subquadratic logistic growth

<p style='text-indent:20px;'>In this paper, we investigate the large time behavior of the generalized solution to the Keller-Segel-Stokes system with logistic growth <inline-formula><tex-math id="M1">\begin{document}$ \rho n-rn^{\alpha } $\end{document}</tex-math></inline-formula> in a bounded domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset \mathbb R^d $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M3">\begin{document}$ (d\in\{2, 3\}) $\end{document}</tex-math></inline-formula>, as given by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{l} &amp;n_t+{{\bf{u}}}\cdot\nabla n = \Delta n-\chi\nabla\cdot\big(n\nabla c\big)+\rho n-rn^{\alpha }, \\ &amp;c_t+{{\bf{u}}}\cdot\nabla c = \Delta c-c+n, \\ &amp;{{\bf{u}}}_t+\nabla P = \Delta{{\bf{u}}}+n\nabla\phi, \\ &amp;\nabla\cdot{{\bf{u}}} = 0 \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>for the unknown <inline-formula><tex-math id="M4">\begin{document}$ (n, c, {{\bf{u}}}, P) $\end{document}</tex-math></inline-formula>, with prescribed and suitably smooth <inline-formula><tex-math id="M5">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. Our result shows that if <inline-formula><tex-math id="M6">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \rho $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ r $\end{document}</tex-math></inline-formula> satisfy</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \alpha &gt; \frac{2d-2}{d}\quad\mathrm{and}\quad\chi^2&lt; K\rho^{ \frac{\alpha -3}{\alpha -1}}r^{ \frac{2}{\alpha -1}} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with some positive constant <inline-formula><tex-math id="M10">\begin{document}$ K $\end{document}</tex-math></inline-formula> depending on <inline-formula><tex-math id="M11">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>, the generalized solution converges to a constant steady state ((<inline-formula><tex-math id="M14">\begin{document}$ \frac{\rho}{r})^{ \frac{1}{\alpha -1}}, ( \frac{\rho}{r})^{ \frac{1}{\alpha -1}}, {\bf 0} $\end{document}</tex-math></inline-formula>) after a large time. Our proof is based on the decay property of a functional involving <inline-formula><tex-math id="M15">\begin{document}$ n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M16">\begin{document}$ c $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ {\bf{u}} $\end{document}</tex-math></inline-formula>.</p>