Global bounded weak solutions and asymptotic behavior to a chemotaxis-Stokes model with non-Newtonian filtration slow diffusion

In this paper, we deal with the following chemotaxis-Stokes model with non-Newtonian filtration slow diffusion (namely, p > 2) {n(t)+ u center dot del n = del center dot (|del n(|p-2)del n) - chi del center dot (n del c), c(t) + u center dot del c - Delta c = -cn, u(t) + del pi = Delta u + n del phi, divu = 0 in a bounded domain Omega of R-3 with zero-flux boundary conditions and no-slip boundary condition. Similar to the study for the chemotaxis-Stokes system with porous medium diffusion, it is also a challenging problem to find an optimal p-value (p >= 2) which ensures that the solution is global bounded. In particular, the closer the value of pis to 2, the more difficult the study becomes. In the present paper, we prove that global bounded weak solutions exist whenever p > p* (approximate to 2.012). It improved the result of [21,22], in which, the authors established the global bounded solutions for p > 23/11. Moreover, we also consider the large time behavior of solutions, and show that the weak solutions will converge to the spatially homogeneous steady state ((n) over bar (0), 0, 0). Comparing with the chemotaxis-fluid system with porous medium diffusion, the present convergence of nis proved in the sense of L-infinity-norm, not only in L-p-norm or weak-* topology. (c) 2021 Elsevier Inc. All rights reserved.