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

Abstract This paper deals with the following quasilinear chemotaxis-growth system u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , x ∈ Ω , t > 0 , v t = Δ v − v + w , x ∈ Ω , t > 0 , τ w t + δ w = u , x ∈ Ω , t > 0 , in a smoothly bounded domain Ω ⊂ R n ( 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 ) ≥ D 0 u θ with D 0 > 0 and θ ∈ R . Under the assumption θ > 1 − 4 n , it is proved that whenever μ > 0 , τ > 0 and δ > 0 , for any given nonnegative and suitably smooth initial data ( u 0 , v 0 , w 0 ) satisfying u 0 ≢ 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 ) | | W 1 , s ( Ω ) with s ∈ [ 1 , 2 n n − 2 ) to estimate the boundedness of | | ∇ v ( ⋅ , t ) | | L 2 q ( 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 .