带对数趋化敏感度的Keller-Segel模型的定性研究

Chemotaxis has an important theoretical and practical significance in biology and medicine, and the qualitative analysis of it can help us understand chemotaxis phenomenon deeply. Here we are considering the Keller-Segel model with logarithmic sensitivity, which is a very famous chemotaxis model. Compared with the classical simplified Keller-Segel model, this model not only contains a cross-diffusion term, but also has singularity, which causes great difficulties for the mathematical study. Now the research works on this model are mainly about the nonlinear stability of travelling waves and the well-posedness of large perturbation around constant states. However, the nonlinear stability of large perturbation around the travelling waves and the existence and stability of self-similar solutions are still unresolved. The applicant has been devoted to the study of the nonlinear evolution equations and chemotaxis models, and has also done related works on the stability of travelling waves and large time behavior of solutions. Therefore, we plan to investigate the fore unresolved problems. The significance of this research is that it helps us to understand more about the asymptotic behavior of solutions, and meanwhile, through the study on the self-similar solutions, we can also understand the singularity mechanism more deeply, thus providing a new idea to study the general Cauchy problem.

趋化性在生物学和医学上具有很重要的理论和现实意义，对它的定性分析有助于我们更加深入地了解趋化现象。我们要考虑的就是一类很著名的趋化模型，即带对数趋化敏感度的Keller-Segel模型。此模型与经典的简化Keller-Segel模型相比，除了含有交错扩散项以外，还具有奇性，这一点为数学研究带来了巨大困难。目前关于这个方程的研究工作主要是集中于行波解的非线性稳定性和常状态附近大扰动解的适定性这两个方面。然而大初始扰动下行波解的非线性稳定性问题以及自相似解的存在性和稳定性问题仍亟待解决。申请人一直致力于非线性发展方程和趋化模型的学习与研究，在行波解的稳定性和解的大时间行为方面也做过相关工作，因此决定以上述两个问题为研究内容。该研究的意义在于一方面有助于我们进一步了解解的渐进行为，同时通过对自相似解的研究也可以让我们更加深入地理解奇性项的作用机制，从而为一般初值柯西问题的研究提供新的思路。

趋化性在生物学和医学上有着重要的理论和实际意义, 是近年来的一个热门研究领域。本项目研究的就是一类著名的趋化模型——带对数趋化敏感度的Keller-Segel（KS）模型，旨在研究其大初始扰动下行波解的非线性稳定性问题以及自相似解的存在性和稳定性问题。经过研究，我们分别得到了1维及2维情形下径向对称的非负自相似解存在的必要性条件和充分性条件，以及自相似解在径向上的单调性和衰减性。此外，我们也研究了具有大质量初值的经典KS模型，给出了大质量整体经典解存在的一个充分性条件；对于两维抛物-抛物-椭圆型吸引-排斥趋化模型的Cauchy问题，我们证明了当排斥效应强于或等于聚集效应，或者当排斥效应弱于聚集效应但初始质量小于“临界值”时经典解的整体存在性。上述结果有助于我们进一步了解解的渐近行为和趋化模式。