Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system
Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system
复制标题
Keller-Segel 型反应扩散系统中信号依赖性运动的影响
DOI：
10.1142/s0218202517500282
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发表时间：
2017
影响因子：
3.5
通讯作者：
Michael Winkler
中科院分区：
文献类型：
--
作者：
Youshan Tao;Michael Winkler
本文考虑Keller-Segel型抛物方程组 ut = Δ（u ∈（v）），x ∈ Ω，t > 0，<？A3B2 [rowsep=4pt]？> vt = Δv − v + u，x ∈ Ω，t > 0，（？）在光滑有界凸区域中，在无通量边界条件下，它最近被提出作为通过所谓的“自陷”机制形成条纹图案过程的模型。在二维的情况下，在形成鲜明对比的经典Keller-Segel模型中，大数据的解决方案可能会在有限时间内爆炸，所有适当的正规初始数据的相关的初始值问题被认为是拥有一个全球定义的有界经典解决方案，提供的运动功能是一致的积极的。在相应的高维设置，它表明，某些弱解存在全球性的，在特定的三维情况下，这个解决方案实际上是有界的和经典的，如果初始数据是适当的小范数。最后，如果仍然但仅仅是物理上可解释的量是适当小的，则上述弱解最终是光滑的和有界的。
This work considers the Keller–Segel-type parabolic system ut = Δ(uϕ(v)), x ∈ Ω,t > 0, <?A3B2 [rowsep=4pt]?>vt = Δv − v + u,x ∈ Ω,t > 0, (⋆) in a smoothly bounded convex domain,, under no-flux boundary conditions, which has recently been proposed as a model for processes of stripe pattern formation via so-called “self-trapping” mechanisms. In the two-dimensional case, in stark contrast to the classical Keller–Segel model in which large-data solutions may blow up in finite time, for all suitably regular initial data the associated initial value problem is seen to possess a globally-defined bounded classical solution, provided that the motility functionis uniformly positive. In the corresponding higher-dimensional setting, it is shown that certain weak solutions exist globally, where in the particular three-dimensional case this solution actually is bounded and classical if the initial data are suitably small in the norm of. Finally, if stillbut merely the physically interpretable quantityis appropriately small, then the above-weak solutions are proved to become eventually smooth and bounded.