Asymptotic behavior in a quasilinear chemotaxis-growth system with indirect signal production

Abstract

We consider a quasilinear chemotaxis system involving logistic source{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)+μ(u−uγ),x∈Ω,t>0,vt=Δv−v+w,x∈Ω,t>0,wt=Δw−w+u,x∈Ω,t>0, with nonnegative initial data under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn(n⩾1). Here, constants μ>0, γ>1, and D, S are smooth functions fulfilling D(s)⩾K0(s+1)α, |S(s)|⩽K1s(s+1)β−1 for all s⩾0 with α,β∈R and K0,K1>0. Then, if β⩽γ−1, the nonnegative classical solution (u,v,w) is global in time and bounded. Moreover, if μ>0 is sufficiently large, this global bounded solution with nonnegative initial data (u0,v0,w0) satisfies‖u(⋅,t)−1‖L∞(Ω)+‖v(⋅,t)−1‖L∞(Ω)+‖w(⋅,t)−1‖L∞(Ω)→0 as t→∞.