Global solvability and stability to a nutrient-taxis model with porous medium slow diffusion

Abstract: In this paper, we study a nutrient-taxis model with porous medium slow diffusion \begin{align*} \left{ \begin{aligned} &u_t=\Delta u^m-\chi\nabla\cdot(u\nabla v)+\xi uv-\rho u, \ &v_t-\Delta v=-vu+\mu v(1-v), \end{aligned}\right. \end{align*} in a bounded domain $\Omega\subset \mathbb R^3$ with zero-flux boundary condition. It is shown that for any $m>\frac{11}4-\sqrt 3$,the problem admits a global weak solution for any large initial datum. We divide the study into three cases,(i) $\xi\mu=0, \rho\ge 0$; (ii) $\xi\mu\rho>0$; (iii) $\xi\mu>0$, $\rho=0$. In particular, for Case (i) and Case (ii), the global solutions are uniformly bounded. Subsequently, the large time behavior of these global bounded solutions are also discussed. At last, we also extend the results to the coupled chemotaxis-Stokes system. Important progresses for chemotaxis-Stokes system with $m>\frac 76$, $m>\frac 87$ and $m>\frac 98$ have been carried out respectively by \cite{W2, TW2, W3}, but leave a gap for $1<m\le \frac98$. Our result for chemotaxis-Stokes system supplements part of the gap $(\frac{11}4-\sqrt 3, \frac 98)$. Here $\frac{11}4-\sqrt 3\approx 1.018$.