Global boundedness in a Keller-Segel system with flux limitation and logistic source

Abstract

We consider the parabolic chemotaxis system{ut=Δu−∇⋅(uf(|∇v|2)∇v)+ru−μuγ,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, in a smooth bounded domain Ω⊂Rn(n≥1) with the homogeneous Neumann boundary conditions, where r,μ>0, γ>1 and the function f satisfiesf(ξ)=(1+ξ)−α2,for all ξ≥0, with α>0. In the case n≤2, it is shown that the corresponding initial value problem possesses a global bounded classical solution for any α,μ>0. In the case n≥3, if γ=2 and α=n−22n, there exists μ0>0 such that for any μ≥μ0, a global bounded classical solution exists.