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

Abstract

In this paper, we deal with the following chemotaxis-Stokes model with non-Newtonian filtration slow diffusion (namely, p>2){nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−χ∇⋅(n∇c),ct+u⋅∇c−Δc=−cn,ut+∇π=Δu+n∇φ,divu=0 in a bounded domain Ω of R3 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 p is to 2, the more difficult the study becomes. In the present paper, we prove that global bounded weak solutions exist wheneverp>p⁎(≈2.012). It improved the result of [21,22], in which, the authors established the global bounded solutions for p>2311. 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‾0,0,0). Comparing with the chemotaxis-fluid system with porous medium diffusion, the present convergence of n is proved in the sense of L∞-norm, not only in Lp-norm or weak-* topology.