A new result for boundedness in the quasilinear parabolic–parabolic Keller–Segel model (with logistic source)

Abstract

The current paper considers the boundedness of solutions to the following quasilinear Keller–Segel model (with logistic source) (KS)ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+μ(u−u2),x∈Ω,t>0,vt−Δv=u−v,x∈Ω,t>0,(D(u)∇u−χu⋅∇v)⋅ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω,where Ω⊂RN(N≥1) is a bounded domain with smooth boundary ∂Ω,χ>0 and μ≥0. One novelty of this paper is that we find a new a-priori estimate ∫Ωuχmax{1,λ0}(χmax{1,λ0}−μ)+−ε(x,t)dx, so that, we develop newLp-estimate techniques and thereby obtain the boundedness results, where CGN and λ0≔λ0(γ) are the constants which are corresponding to the Gagliardo–Nirenberg inequality (see Lemma 2.2 ) and the maximal Sobolev regularity (see Lemma 2.3). To our best knowledge, this seems to be the first rigorous mathematical result which indicates the relationship between m and μχ that yields the boundedness of the solutions, where m is the exponent of diffusion term D(u). The above-mentioned results have significantly improved and extended previous results of several authors.