Asymptotic behavior of a quasilinear Keller–Segel system with signal-suppressed motility

This paper is concerned with the density-suppressed motility model: ut=Δ(umvα)+βuf(w),vt=DΔv-v+u,wt=Δw-uf(w) in a smoothly bounded convex domain Ω ⊂ R², where m> 1 , α> 0 , β> 0 and D> 0 are parameters, the response function f satisfies f∈ C¹([0 , ∞)) , f(0) = 0 , f(w) > 0 in (0 , ∞). This system describes the density-suppressed motility of Eeshcrichia coli cells in the process of spatio-temporal pattern formation via so-called self-trapping mechanisms. Based on the duality argument, it is shown that for suitable large D the problem admits at least one global weak solution (u, v, w) which will asymptotically converge to the spatially uniform equilibrium (u¯ + βw¯ , u¯ + βw¯ , 0) with u0¯=1|Ω|∫Ωu(x,0)dx and w0¯=1|Ω|∫Ωw(x,0)dx in L^∞(Ω).