几类Keller-Segel趋化模型解的有界性与渐近行为研究

In this project, we will study nonlinear partial differential equations derived from chemotaxis-related biological fields, including the following two aspects. Firstly, taking into account the effects of nonlinear diffusion and nonlinear chemotactic sensitivity for Keller-Segel chemotaxis models, by means of both energy estimates and Moser iterative technique, we study global boundedness and asymptotic behavior for the parabolic-parabolic chemotaxis model with nonlinear diffusion and nonlinear chemotactic sensitivity; secondly, by using the entropy estimates and contradiction method, we consider boundedness and large time asymptotic behavior for Keller-Segel chemotaxis models with nonlinear rotational sensitivity and consumption of chemoattractant. Through the study of these aspects, we can better understand the background of these issues, in order to achieve a combination of theory and practice, so that they can play important roles in mathematical biology models and other applications. We strive to achieve a breakthrough about these frontier and hot problems in mathematical biology.

本课题拟研究来源于生物趋化相关领域中的非线性偏微分方程组，将从以下两个方面展开：首先，考虑到非线性扩散和非线性趋化敏感函数对Keller-Segel趋化模型的作用，利用能量估计方法和Moser迭代技巧来研究非线性扩散的抛物-抛物趋化模型解的全局有界性与渐近行为；其次，利用熵估计方法和反证法思想来研究具有非线性旋转趋化敏感和趋化信号物质被细胞消耗的Keller-Segel趋化模型解的有界性与大时间渐近行为。通过对上述两个方面的研究，能够更好地了解这些问题的背景，以求达到理论与实践相结合，使这些数学模型能够在生物学等应用中充分发挥作用。力争在这些前沿和热点的生物数学问题中有所突破。

本项目基本上是按原计划进行研究，首先利用先验估计、迭代技巧和能量方法，研究一类带有信号依赖的趋化-增长模型解的全局有界性与渐近行为；其次考虑一类高维情形下抛物-椭圆-ODE趋化-趋触模型解的有界性与衰减速率估计；最后运用最大Sobolev正则性估计和半群技巧，对一类吸引-排斥趋化-增长模型和一类带有两个化学物质与两个种群竞争的趋化模型解的全局有界性问题进行了研究。在这些前沿和热点的生物数学问题中取得了一些研究成果。