Large time behavior in a quasilinear chemotaxis model with indirect signal absorption

Abstract

This paper considers the following chemotaxis system ut=∇⋅D(u)∇u−∇⋅S(u)∇v,x∈Ω,t>0,vt=Δv−vw,x∈Ω,t>0,wt=−δw+u,x∈Ω,t>0,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn(n⩾2), where the parameter δ>0 and D, S are smooth functions satisfying D(u)⩾K0(u+1)α, 0⩽S(u)⩽K1(u+1)β−1u with α,β∈R and K0,K1>0. Suppose that β−α<1n+12, this system admits a global bounded classical solution (u,v,w) fulfilling ‖u⋅t−ū0‖L∞Ω+‖v⋅t−0‖L∞Ω+‖w⋅t−ū0δ‖L∞Ω→0as t→∞, where ū0≔1|Ω|∫Ωu0.