A Novel Geometric Approach to Shocks in Reaction-Nonlinear Diffusion Models

反应-非线性扩散模型在细胞迁移和种群动力学研究中起着至关重要的作用。然而，聚集或反向扩散的存在会导致激波的形成——不同密度的细胞群之间明显而尖锐的界面——并导致模型的崩溃。该项目将开发新的几何方法来解释这些冲击波的形成和时间演变，同时将现有的正则化技术统一在一个单一的几何旗帜下。它将在奇异摄动理论和稳定性分析方面设计创新工具，以确定冲击波产生的关键参数及其动态行为。

Reaction-nonlinear diffusion models play a vital role in the study of cell migration and population dynamics. However, the presence of aggregation, or backward diffusion, leads to the formation of shock waves - distinct, sharp interfaces between different populations of densities of cells - and the breakdown of the model. This project will develop new geometric methods to explain the formation and temporal evolution of these shock waves, while simultaneously unifying existing regularisation techniques under a single, geometric banner. It will devise innovative tools in singular perturbation theory and stability analysis that will identify key parameters in the creation of shock waves, as well as their dynamic behaviour.