Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

Abstract

We study the chemotaxis effect vs. logistic damping on boundedness for the well-known minimal Keller–Segel model with logistic source:{ut=∇⋅(∇u−χu∇v)+u−μu2,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0 in a smooth bounded domain Ω⊂R2 with χ,μ>0, nonnegative initial data u0, v0, and homogeneous Neumann boundary data. It is well known that this model allows only for global and uniform-in-time bounded solutions for any χ,μ>0. Here, we carefully employ a simple and new method to regain its boundedness, with particular attention to how upper bounds of solutions qualitatively depend on χ and μ. More, precisely, it is shown that there exists C=C(u0,v0,Ω)>0 such that‖u(⋅,t)‖L∞(Ω)≤C[1+1μ+χK(χ,μ)N(χ,μ)] and‖v(⋅,t)‖W1,∞(Ω)≤C[1+1μ+χ83μK83(χ,μ)]=:CN(χ,μ) uniformly on [0,∞), whereK(χ,μ)=M(χ,μ)E(χ,μ),M(χ,μ)=1+1μ+χ(1+1μ2) andE(χ,μ)=exp⁡[χCGN22min⁡{1,2χ}(4μ‖u0‖L1(Ω)+132μ2|Ω|+‖∇v0‖L2(Ω)2)]. We notice that these upper bounds are increasing in χ, decreasing in μ, and have only one singularity at μ=0, where the corresponding minimal model (removing the term u−μu2 in the first equation) is widely known to possess blow-ups for large initial data.