A new result for boundedness of solutions to a higher-dimensional quasilinear chemotaxis system with a logistic source

Abstract

We deal with the boundedness of solutions of a class of quasilinear chemotaxis systems generalizing the prototype(⁎){ut=∇⋅(ϕ(u)∇u)−∇⋅(u∇v)+μu(1−u),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω, where RN(N≥2) is a bounded domain with zero-flux boundary condition, μ>0 is a positive parameter, and ϕ(u)=(u+1)−α. It is shown that for all reasonably regular initial data, system (⁎) admits a global existence and boundedness of solutions when α<α0 for some α0>2−NN+2, thereby proving that the value α=2−NN+2 is not critical in this regard. This extends previous results that rely on a different energy-type inequality.