Singular limits of nonlinear elliptic and parabolic PDE

非线性椭圆和抛物线偏微分方程的奇异极限

• 批准号: EP/J011487/1
• 负责人: Elaine Crooks
• 金额: $ 1.18万
• 依托单位: Swansea University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2012
• 资助国家: 英国
• 起止时间: 2012 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FJ011487%2F1
• 关键词: Singular; limits; nonlinear; elliptic; parabolic

Strong-competition limits of competing-population systems with two componentshave proved an effective way of obtaining information about the competition system for large values of the competition parameter. Understanding the dynamics of such systems is very difficult in general, due to their lack of variational structure. On the other hand, the strong-interaction limit is a scalar equation which has a Lyapunov function and is much more easily studied. Such singular limits are both mathematically useful, and correspond to the important biological and physical phenomena of spatial segregation and phase separation.For competition systems with more than two components, however, the techniquesthat are effective in the two-component case fail and new ideas are needed. Some success has been had been recently, using tools from free-boundary problems, to show convergence of solutions of the competition system to a solution of a limitproblem. The central goal of this proposal is to prove results on the so-called converse problem for multi-component systems; that is, on whether or not a solution of the limit problem does in fact arise as the limit of solutions of the original system. Such converse problemsare an essential and important part of the understanding of any singular limit, and have yet to be addressed at all for strong-competition limits of multi-component systems.We also aim to prove results on the converse problem for competition systems oftwo components when the interaction terms are large only on a subdomain, and forsystems with a different, cubic type of competitive coupling, which occurs in modelling certain quantum mechanical systems.

具有两个组成部分的竞争种群系统的强竞争极限已被证明是在竞争参数较大时获取竞争系统信息的有效方法。由于缺乏变分结构，理解此类系统的动力学通常非常困难。另一方面，强相互作用极限是一个标量方程，具有李亚普诺夫函数，更容易研究。这种奇异极限在数学上都是有用的，并且对应于空间分离和相分离的重要生物和物理现象。然而，对于具有两个以上组成部分的竞争系统，在双组成部分情况下有效的技术失败了，新的想法正在出现。需要。最近已经取得了一些成功，使用自由边界问题的工具来展示竞争系统的解决方案与极限问题的解决方案的收敛性。该提案的中心目标是证明多组件系统的逆问题的结果；也就是说，极限问题的解是否实际上作为原始系统解的极限而出现。此类逆问题是理解任何奇异极限的基本且重要的部分，并且对于多分量系统的强竞争极限而言尚未得到解决。我们还旨在证明两分量竞争系统的逆问题的结果当相互作用项仅在子域上较大时，以及对于具有不同立方类型竞争耦合的系统时，这种情况发生在对某些量子力学系统进行建模时。