We study herein the initial–boundary value problem for the flux-limited chemotaxis model with nonlinear signal production ut = ∇⋅ up∇u u2 + |∇u|2 − χ∇⋅ uq∇v 1 + |∇v|2 ,x ∈ Ω,t > 0,0 = Δv − μ(t) + uk, x ∈ Ω,t > 0, subject to no-flux boundary conditions in a ball Ω = BR(0) ⊂ ℝn(n ≥ 1,R > 0), where χ > 0,k > 0,p ≥ 1,q ≥ 1, μ(t) := 1 |Ω|∫Ωuk(⋅,t)dx. For radially symmetric and positive initial data u0 ∈ C3(Ω¯), it is proved that the corresponding solution is globally bounded when p >max{k,q}. This result not only fills up a gap left in [M. Mizukami, T. Ono and T. Yokota, Extensibility criterion ruling out gradient blowup in a quasilinear degenerate chemotaxis system, J. Differ. Equ. 267(2019) 5115–5164; Y. Chiyoda, M. Mizukami and T. Yokota, Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation, Acta. Appl. Math. 167 (2020) 231–259] when k = 1, but also extend the boundedness result from the special case k = 1 to the general case k > 0. Moreover, under the condition 1 ≤ p ≤ qandk > 1, it is shown that if χ is sufficiently large, then there exists initial data u0 ∈ C3(Ω¯) such that the corresponding solution blows up at finite time T∗ in the sense that limt↗T∗sup ∥u(⋅,t)∥L∞(Ω) = ∞. This blow-up result generalizes the works for the linear signal production case in (N. Bellomo and M. Winkler, Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Am. Math. Soc. Ser. B 4 (2017) 31–67; Y. Chiyoda, M. Mizukami and T. Yokota, Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation, Acta. Appl. Math. 167 (2020) 231–259) to the nonlinear case k > 1.

中文翻译：

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有限通量趋化模型中非线性信号产生对有界性和有限时间爆炸的影响

我们在这里研究具有非线性信号产生的通量限制趋化模型的初始边界值问题 你吨 = ∇⋅ 你p∇你 你2 + |∇你|2 - χ∇⋅ 你q∇v 1 + |∇v|2 ,X ∈ Ω,吨 > 0,0 = Δv - μ(吨) + 你ķ, X ∈ Ω,吨 > 0, 受球中无通量边界条件的影响Ω = 乙R(0) ⊂ ℝn(n ≥ 1,R > 0)， 在哪里χ > 0,ķ > 0,p ≥ 1,q ≥ 1,μ(吨) ：= 1 |Ω|∫Ω你ķ(⋅,吨)dX. 对于径向对称和正初始数据你0 ∈ C3(Ω¯)，证明了对应的解是全局有界的，当p >最大限度{ķ,q}. 这一结果不仅填补了[M. Mizukami，T. Ono 和 T. Yokota，可扩展性准则排除准线性简并趋化系统中的梯度爆炸，J. 不同。等。 267(2019) 5115–5164；Y. Chiyoda、M. Mizukami 和 T. Yokota，具有通量限制的拟线性简并趋化系统中的有限时间爆炸，学报。应用程序。数学。 167(2020) 231–259] 当ķ = 1, 还扩展了特例的有界结果ķ = 1一般情况ķ > 0. 而且，条件下 1 ≤ p ≤ q和ķ > 1, 表明如果χ足够大，则存在初始数据你0 ∈ C3(Ω¯)使得相应的解决方案在有限时间内爆炸吨*在某种意义上说 林吨↗吨*支持 ∥你(⋅,吨)∥大号∞(Ω) = ∞. 这个爆破结果概括了线性信号产生案例的工作（N. Bellomo 和 M. Winkler，在具有通量限制的退化趋化系统中的有限时间爆破，反式。是。数学。社会党。塞尔.乙 4（2017）31-67；Y. Chiyoda、M. Mizukami 和 T. Yokota，具有通量限制的拟线性简并趋化系统中的有限时间爆炸，学报。应用程序。数学。 167(2020) 231–259) 到非线性情况ķ > 1.