Boundedness in the higher-dimensional quasilinear chemotaxis-growth system with indirect attractant production

This paper deals with the following quasilinear chemotaxis-growth system ut=∇⋅(D(u)∇u)−∇⋅(u∇v)+μu(1−u),x∈Ω,t>0,vt=Δv−v+w,x∈Ω,t>0,τwt+δw=u,x∈Ω,t>0,in a smoothly bounded domain Ω⊂Rn(n≥3) under zero-flux boundary conditions. The parameters μ,δ and τ are positive and the diffusion function D(u) is supposed to generalize the prototype D(u)≥D0uθ with D0>0 and θ∈R. Under the assumption θ>1−4n, it is proved that whenever μ>0, τ>0 and δ>0, for any given nonnegative and suitably smooth initial data (u0, v0, w0) satisfying u0≢0, the corresponding initial-boundary problem possesses a unique global solution which is uniformly-in-time bounded. The novelty of the paper is that we use the boundedness of the ||v(⋅,t)||W1,s(Ω) with s∈[1,2nn−2) to estimate the boundedness of ||∇v(⋅,t)||L2q(Ω)(q>1). Moreover, the result in this paper can be regarded as an extension of a previous consequence on global existence of solutions by Hu et al. (2016) under the condition that D(u)≡1 and n=3.