Asymptotic stabilization for a class of chemotaxis-consumption systems with generalized logistic source

Abstract

This paper is concerned with a chemotaxis-consumption system ut=∇⋅(∇u−uS(x,u,v)⋅∇v)+ρu−μul,vt=Δv−uv,under no-flux boundary conditions in a smooth bounded domain Ω⊂Rn(n⩾1), where the chemotactic sensitivity tensor S∈C2(Ω¯×[0,+∞)2;Rn×n) fulfills that there exists some nondecreasing function S0 on [0,+∞) such that |S(x,u,v)|⩽S0(v)for all(x,u,v)∈Ω¯×[0,+∞)×[0,+∞).We show that for any ρ,μ>0 and l>1, any generalized solution of the above system asymptotically approaches to the nontrivial spatially homogeneous steady state ρμ1l−1,0as t→+∞.