Boundedness and large time behavior for a chemotaxis system with signal-dependent motility and indirect signal consumption. (April 2022) - Li, Dan; Li, Zhongping; Zhao, Jie
Abstract: In this paper, we study the following chemotaxis system with signal-dependent motility, indirect signal consumption and logistic source u t = Δ ( u γ ( v ) ) + ρ u − μ u l, x ∈ Ω, t > 0, v t = Δ v − v w, x ∈ Ω, t > 0, w t = − δ w + u, x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n, where the motility function γ ( v ) ∈ C 3 ( ( 0, + ∞ ) ), γ ( v ) > 0, γ ′ ( v ) < 0 on ( 0, + ∞ ), lim v → ∞ γ ( v ) = 0, ρ > 0, μ > 0, l > 1 and δ > 0 . The purpose of this paper is to prove that if l > m a x { 1, n 2 }, then the system possesses a global solution. In addition, if l satisfies l ≥ 2, if  n ≤ 3, > n 2, if  n ≥ 4, then the solution ( u, v, w ) satisfies ‖ u ( ⋅, t ) − ( ρ μ ) 1 l − 1 ‖ L ∞ ( Ω ) + ‖ v ( ⋅, t ) ‖ L ∞ ( Ω ) + ‖ w ( ⋅, t ) − 1 δ ( ρ μ ) 1 l − 1 ‖ L ∞ ( Ω ) → 0 a s t → ∞ .