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

Abstract

This paper is concerned with the density-suppressed motility model:, where m > 1, α > 0, β > 0 and D > 0 are parameters, the response functionThis system describes the density-suppressed motility of Eeshcrichia coli cells in 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 0 + βw 0 , u 0 + βw 0 , 0) with u 0 = 1 |Ω| Ω u(x, 0)dx and w 0 = 1 |Ω| Ω w(x, 0)dx in L ∞ (Ω).