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This paper is devoted to studying the following quasilinear parabolic-elliptic-elliptic chemotaxis system { u t = ∇ ⋅ ( φ ( u ) ∇ u − ψ ( u ) ∇ v ) + a u − b u γ , x ∈ Ω , t > 0 , 0 = Δ v − v + w γ 1 , x ∈ Ω , t > 0 , 0 = Δ w − w + u γ 2 , x ∈ Ω , t > 0 , with homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ R n ( n ≥ 1 ) , where a , b , γ 2 > 0 , γ 1 ≥ 1 , γ > 1 and the functions φ , ψ ∈ C 2 ( [ 0 , ∞ ) satisfy φ ( s ) ≥ a 0 ( s + 1 ) α and | ψ ( s ) | ≤ b 0 s ( 1 + s ) β − 1 for all s ≥ 0 with a 0 , b 0 > 0 and α , β ∈ R . It is proved that if γ − β ≥ γ 1 γ 2 , the classical solution of system would be globally bounded. Furthermore, a specific model for γ 1 = 1 , γ 2 = κ and γ = κ + 1 with κ > 0 is considered. If β ≤ 1 and b > 0 is large enough, there exist C κ , μ 1 , μ 2 > 0 such that the solution ( u , v , w ) satisfies ‖ u ( ⋅ , t ) − ( b a ) 1 κ ‖ L ∞ ( Ω ) + ‖ v ( ⋅ , t ) − b a ‖ L ∞ ( Ω ) + ‖ w ( ⋅ , t ) − b a ‖ L ∞ ( Ω ) ≤ { C κ e − μ 1 t , if κ ∈ ( 0 , 1 ] , C κ e − μ 2 t , if κ ∈ ( 1 , ∞ ) , for all t ≥ 0. The above results generalize some existing results.
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