LAUSR

Abstract This work considers a chemotaxis system with signal-dependent motility in a smooth bounded domain Ω ⊂ R n . If λ ∈ R , μ > 0 and l > max ⁡ { n + 2 4 , 1 } are constants, then the system { u t = Δ ( γ ( v ) u ) + λ u − μ u l , x ∈ Ω , t > 0 , v t = Δ v − v + w , x ∈ Ω , t > 0 , w t = Δ w − w + u , x ∈ Ω , t > 0 , with homogeneous Neumann boundary conditions possesses a global solution. Moreover, the solution satisfies ‖ u ( ⋅ , t ) − ( λ + μ ) 1 l − 1 ‖ L ∞ + ‖ v ( ⋅ , t ) − ( λ + μ ) 1 l − 1 ‖ L ∞ + ‖ w ( ⋅ , t ) − ( λ + μ ) 1 l − 1 ‖ L ∞ → 0 as t → ∞ under some extra hypotheses, where λ + = max ⁡ { λ , 0 } .