BOUNDEDNESS AND LARGE TIME BEHAVIOR IN A TWO-DIMENSIONAL KELLER-SEGEL-NAVIER-STOKES SYSTEM WITH SIGNAL-DEPENDENT DIFFUSION AND SENSITIVITY

This paper is concerned with the following Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity {n(t) + u . del n = (d(c)del n) - del(chi(c)n del c) + an - bn2, x is an element of Omega, t > 0, c(t) + u . del c + n - c, 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, (*) in a bounded smooth domain Omega subset of R-2 with homogeneous Neumann boundary conditions, where a >= 0 and b > 0 are constants, and the functions d(c) and chi(c) satisfy the following assumptions: (d(c),x(c)) is an element of [C-2([0, infinity))](2) with d(c),x(c) > 0 for all c >= 0, d'(c) < 0 and lim(c) (->) (infinity) d(c) = 0 lim(c -> infinity) chi(c)/d(c) and lim(c -> infinity) d(c) exist. The difficulty in analysis of system (*) is the possible degeneracy of diffusion due to the condition lim(c -> infinity) d(c) = 0. In this paper, we will use function d(c) as weight function and employ the method of energy estimate to establish the global existence of classical solutions of (*) with uniform-in-time bound. Furthermore, by constructing a Lyapunov functional, we show that the global classical solution (n, e, u) will converge to the constant state (a/b, a/b/ 0) if b > K-0/16 with K-0 = max(0 <= c <=infinity) vertical bar chi(c vertical bar)(2)/d(c).