A NOTE TO THE LARGE-TIME BEHAVIOR OF A 3D CHEMOTAXIS-NAVIER-STOKES SYSTEM WITH POROUS MEDIUM SLOW DIFFUSION

In this note, we consider the large time behavior of the following chemotaxis-Navier-Stokes system {n(t) + u . del n = Delta n(m) - del . (n del(c)), x is an element of Omega, t > 0, c(t) + u . del(c) = Delta c - nc, x is an element of Omega, t > 0, u(t) + u del u = Delta u + del P + n del phi, x is an element of Omega, t > 0, del . u = 0, x is an element of Omega, t > 0 with m > 1 in spatially three-dimensional setting. The global weak solution (n, c, u) to the no-flux/no-flux/no-slip initial-boundary value problem has been constructed by Zhang and Li (J. Differential Equations, 2015). Here, we will show that such a weak solution will stabilize to the constant equilibrium ((n(0)) over bar, 0, 0) with (n(0)) over bar = 1/vertical bar Omega vertical bar integral(Omega) n(0) as t -> infinity.