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; 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.

具有两个组件的竞争竞争系统的强竞争限制被证明是获取有关大量竞争参数的竞争系统信息的有效方法。一般而言，由于缺乏变化结构，理解这种系统的动态非常困难。另一方面，强相互作用极限是标量方程，具有Lyapunov函数，并且更容易研究。这种奇异的极限在数学上都是有用的，并且对应于空间隔离和相位分离的重要生物学和物理现象。对于具有两个以上组成部分的竞争系统，技术在两组式案例中是有效的。最近，使用了自由边界问题的工具来显示竞争系统解决方案与极限问题解决方案的收敛性。该提案的核心目标是证明有关多组分系统所谓的匡威问题的结果；也就是说，关于限制问题的解决方案是否确实是作为原始系统解决方案的极限而出现的。这种相反的问题是对任何奇异极限的理解的重要和重要部分，并且对于多组分系统的强大竞争极限，尚未解决。我们还旨在证明竞争系统的相互作用问题的结果是，当交互作用的相互作用术语仅在一个子域中和类型的竞争性的竞争性couples的模型中，相互作用术语很大，并且在竞争性的模型中，该系统具有不同的模型。