主题：Asymptotic stability of spiky steady states for a singular chemotaxis model with signal-suppressed motility (具有信号抑制运动的奇异趋化模型的尖峰解的渐近稳定性)

主讲人：东北师范大学 李敬宇教授

主持人：数学学院 林可副教授

时间：2022年11月09日（周三）10:00-11:00

直播平台及会议ID：腾讯会议，666-358-132

主办单位：数学学院 科研处

主讲人简介：

李敬宇，东北师范大学教授，主要研究方向是偏微分方程。已在 Proc. London Math. Soc., SIAM J. Math. Anal., Math. Models Methods Appl. Sci., J. Differential Equations等期刊发表多篇论文，获得多项国家和省级基金的资助。

内容提要：

We study the nonlinear stability of spiky solutions to a chemotaxis model of consumption type with singular signal-suppressed motility in the half space. We show that, when the no-flux boundary condition for the bacteria density and the nonhomogeneous Dirichlet boundary condition for the nutrient are prescribed, this chemotaxis model admits a unique smooth spiky steady state, and it is nonlinearly stable under appropriate perturbations. The challenge of the problem is that there are two types of singularities involved in the model: one is the logarithmic singularity of the sensitive function; and the other is the inverse square singularity of the motility. We employ a Cole-Hopf transformation to relegate the former singularity to a nonlocality that can be resolved by the method of anti-derivative. To deal with the latter singularity, we construct an approximate system that retains a key structure of the original singular system in the local theory, and develop a new strategy, which combines a weighted elliptic estimate and the weighted energy estimate, to establish a priori estimate in the global theory.

我们研究了半空间中具有奇异信号抑制运动的耗散型趋化性模型的尖峰解的非线性稳定性。结果表明当给定了细菌密度的无通量边界条件和化学信号物质的非均匀狄利克雷边界条件时，该趋化模型具有唯一的光滑尖峰解，并且在适当的扰动下是非线性稳定的。该问题的挑战在于该模型中涉及两种类型的奇点：一种是灵敏度函数的对数奇点;另一个是运动的平方反比奇点。我们采用Cole-Hopf变换将第一种奇点化为可以通过反导数方法解决的非局部性问题。为了处理第二种奇点，我们首先构建了一个近似系统，该系统保留了局部理论中原奇异系统的关键结构。再次，将加权椭圆估计和加权能量估计相结合建立了全局理论中的先验估计。