论文标题
与局部传感的趋化模型的延迟爆破
Delayed Blow-Up for Chemotaxis Models with Local Sensing
论文作者
论文摘要
本文的目的是分析基于局部感应机制的趋化模型，而不是著名的最小凯勒 - 塞格模型中使用的梯度感应机制。我们研究的模型具有与最小凯勒 - 塞格模型相同的熵，但动力学不同，可以最大程度地减少此熵。因此，存在固定溶液或爆炸的质量条件是相同的，但是我们使有趣的观察结果是，使用局部感应机制，在超临界质量的情况下，爆炸延迟到无限时间。我们的观察结果是通过数学上严格的，这证明了全球存在的弱解决方案，用于任意大质量和空间维度。我们的模型与最小凯勒 - 塞格模型的关键区别在于，方程的结构允许二元估计值，这意味着在$（h^1）'$ - $ - $ - $ - n-Norm-Norm-Norm-Norm-Norm-Norm-Norm-Norm-Norm中进行的偶性估计，该估计只能在及时使用方形法律生长。额外的$（h^1）'$ - 绑定意味着熵上的下限，这与最小的凯勒 - 塞格模型形成鲜明对比，在超临界情况下，它与下面无限。此外，还研究了解决方案的规律性和独特性。
The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller-Segel model. The model we study has the same entropy as the minimal Keller-Segel model, but a different dynamics to minimize this entropy. Consequently, the conditions on the mass for the existence of stationary solutions or blow-up are the same, however we make the interesting observation that with the local sensing mechanism the blow-up in the case of supercritical mass is delayed to infinite time. Our observation is made rigorous from a mathematical point via a proof of global existence of weak solutions for arbitrary large masses and space dimension. The key difference of our model to the minimal Keller-Segel model is that the structure of the equation allows for a duality estimate that implies a bound on the $(H^1)'$-norm of the solutions, which can only grow with a square-root law in time. This additional $(H^1)'$-bound implies a lower bound on the entropy, which contrasts markedly with the minimal Keller-Segel model for which it is unbounded from below in the supercritical case. Besides, regularity and uniqueness of solutions are also studied.
