The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties
The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties
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DOI：
10.1142/s0218202516500238
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发表时间：
2016-02
影响因子：
3.5
通讯作者：
M. Winkler
中科院分区：
文献类型：
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作者：
M. Winkler
我们考虑趋化性系统ut = Δu − u v v，vt = Δv − uv，最初是由凯勒和西格尔在1971年的第二部开创性著作中提出的。该系统构成了一个原型模型的出租车驱动的模式形成和前传播在各种生物背景下，如肿瘤血管生成，但在高维背景下，任何全球存在的大数据解决方案的理论还缺乏。本文证明了在具有光滑边界的有界平面区域Ω中，对所有合理正则的初值u 0 ≥ 0和v0 > 0，相应的Neumann初边值问题具有整体广义解.因此，特别是解决任意大的初始数据，这超出了以前获得的结果断言全球存在的解决方案，只有在空间一维问题，或在某些小的条件下的初始数据。这个结果的推导是基于时空L2空间中的量的先验估计，其中进一步的有界性和紧致性是通过依赖于平面空间设置使用相关的Moser-Trudinger不等式从前者导出的。此外，还导出了一些进一步的有界性和松弛性质，阿利亚是表明对于任何这样的解，当对所有有限p > 1，t →∞时，Lp（Ω）中有v（n，t）→ 0，并且在适当的广义意义下，量u和n ln v最终分别进入Lp（Ω）和L2（Ω）中的有界集，直径仅由总种群大小n Ω u 0确定。最后，一些数值实验说明了解析得到的结果。
We consider the chemotaxis system ut = Δu −∇⋅u v∇v, vt = Δv − uv, as originally introduced in 1971 by Keller and Segel in the second of their seminal works. This system constitutes a prototypical model for taxis-driven pattern formation and front propagation in various biological contexts such as tumor angiogenesis, but in the higher-dimensional context any global existence theory for large-data solutions is yet lacking. In this work it is shown that in bounded planar domains Ω with smooth boundary, for all reasonably regular initial data u0 ≥ 0 and v0 > 0, the corresponding Neumann initial-boundary value problem possesses a global generalized solution. Thus particularly addressing arbitrarily large initial data, this goes beyond previously gained results asserting global existence of solutions only in spatial one-dimensional problems, or under certain smallness conditions on the initial data. The derivation of this result is based on a priori estimates for the quantities ∇ln(u + 1) and ∇v in spatio-temporal L2 spaces, where further boundedness and compactness properties are derived from the former by relying on the planar spatial setting in using an associated Moser–Trudinger inequality. Furthermore, some further boundedness and relaxation properties are derived, inter alia indicating that for any such solution we have v(⋅,t) → 0 in Lp(Ω) as t →∞ for all finite p > 1, and that in an appropriate generalized sense the quantities u and ∇ln v eventually enter bounded sets in Lp(Ω) and L2(Ω), respectively, with diameters only determined by the total population size ∫Ωu0. Finally, some numerical experiments illustrate the analytically obtained results.