101. Global bounded weak solutions and asymptotic behavior to a chemotaxis-Stokes model with non-Newtonian filtration slow diffusion
- Author
-
Chunhua Jin
- Subjects
Steady state ,Applied Mathematics ,010102 general mathematics ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Bounded function ,Norm (mathematics) ,Convergence (routing) ,Filtration (mathematics) ,Boundary value problem ,0101 mathematics ,Diffusion (business) ,Analysis ,Mathematical physics ,Mathematics - Abstract
In this paper, we deal with the following chemotaxis-Stokes model with non-Newtonian filtration slow diffusion (namely, p > 2 ) { n t + u ⋅ ∇ n = ∇ ⋅ ( | ∇ n | p − 2 ∇ n ) − χ ∇ ⋅ ( n ∇ c ) , c t + u ⋅ ∇ c − Δ c = − c n , u t + ∇ π = Δ u + n ∇ φ , div u = 0 in a bounded domain Ω 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 p is to 2, the more difficult the study becomes. In the present paper, we prove that global bounded weak solutions exist whenever p > p ⁎ ( ≈ 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 ‾ 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 L p -norm or weak-* topology.
- Published
- 2021
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