Abstract We consider a quasilinear chemotaxis system involving logistic source { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) + μ ( u − u γ ) , 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 nonnegative initial data under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n ( n ⩾ 1 ) . Here, constants μ > 0 , γ > 1 , and D, S are smooth functions fulfilling D ( s ) ⩾ K 0 ( s + 1 ) α , | S ( s ) | ⩽ K 1 s ( s + 1 ) β − 1 for all s ⩾ 0 with α , β ∈ R and K 0 , K 1 > 0 . Then, if β ⩽ γ − 1 , the nonnegative classical solution ( u , v , w ) is global in time and bounded. Moreover, if μ > 0 is sufficiently large, this global bounded solution with nonnegative initial data ( u 0 , v 0 , w 0 ) satisfies ‖ u ( ⋅ , t ) − 1 ‖ L ∞ ( Ω ) + ‖ v ( ⋅ , t ) − 1 ‖ L ∞ ( Ω ) + ‖ w ( ⋅ , t ) − 1 ‖ L ∞ ( Ω ) → 0 as t → ∞ .

中文翻译：

---

具有间接信号产生的拟线性趋化-生长系统中的渐近行为

摘要 我们考虑了一个准线性趋化系统，包括逻辑源 { ut = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) + μ ( u − u γ ) , x ∈ Ω , t > 0 , vt = Δ v − v + w , x ∈ Ω , t > 0 , wt = Δ w − w + u , x ∈ Ω , t > 0 ，在光滑有界域Ω中齐次诺依曼边界条件下的非负初始数据⊂ R n ( n ⩾ 1 ) 。这里，常数 μ > 0 , γ > 1 和 D, S 是满足 D ( s ) ⩾ K 0 ( s + 1 ) α , |的光滑函数 S(s) | ⩽ K 1 s ( s + 1 ) β − 1 对于所有 s ⩾ 0 且 α , β ∈ R 和 K 0 , K 1 > 0 。然后，如果 β ⩽ γ − 1 ，则非负经典解 ( u , v , w ) 在时间上是全局的且有界的。此外，如果 μ > 0 足够大，这个具有非负初始数据 (u 0 , v 0 , w 0 ) 的全局有界解满足 ‖ u ( ⋅ , t ) − 1 ‖ L ∞ ( Ω ) + ‖ v ( ⋅ ,