Abstract This paper deals with the chemotaxis-growth system: u t = Δ u − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , v t = Δ v − v + w , τ w t + δ w = u in a smooth bounded domain Ω ⊂ R 3 with zero-flux boundary conditions, where μ, δ, and τ are given positive parameters. It is shown that the solution ( u , v , w ) exponentially stabilizes to the constant stationary solution ( 1 , 1 δ , 1 δ ) in the norm of L ∞ ( Ω ) as t → ∞ provided that μ > 0 and any given nonnegative and suitably smooth initial data ( u 0 , v 0 , w 0 ) fulfills u 0 ≢ 0 , which extends the condition μ > 1 8 δ 2 in [8] .

中文翻译：

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具有间接引诱剂生产的三维趋化生长模型的稳定性

摘要 本文讨论趋化-生长系统： ut = Δ u − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , vt = Δ v − v + w , τ wt + δ w = u in a平滑有界域 Ω ⊂ R 3 具有零通量边界条件，其中 μ、δ 和 τ 为正参数。结果表明，解 ( u , v , w ) 在 L ∞ ( Ω ) 的范数中随着 t → ∞ 指数地稳定为常数平稳解 ( 1 , 1 δ , 1 δ )，条件是 μ > 0 并且任何给定的非负且适当平滑的初始数据 (u 0 , v 0 , w 0 ) 满足 u 0 ≢ 0 ，这扩展了 [8] 中的条件 μ > 1 8 δ 2 。