Higher Dimensional Bubble Profiles in a Sharp Interface Limit of the FitzHugh--Nagumo System
Higher Dimensional Bubble Profiles in a Sharp Interface Limit of the FitzHugh--Nagumo System
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FitzHugh--Nagumo 系统锐界面极限中的更高维气泡轮廓
DOI：
10.1137/17m1144933
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发表时间：
2018
影响因子：
2
通讯作者：
Ren, Xiaofeng
中科院分区：
文献类型：
--
作者：
Chen, Chao-Nien;Choi, Yung-Sze;Hu, Yeyao;Ren, Xiaofeng
Fitzhugh-Nagumo系统产生了定义在区域子集上的非局部几何变分问题。子集的能量包含三个项：其周长、体积和长程自作用项，由屏蔽泊松方程的解的积分表示。当区域是整个空间时，气泡轮廓是球状的固定集。如果空间尺寸为三个或更高，则根据问题的参数，可以有零个、一个或两个气泡轮廓。这与早先的二维空间结果形成对比，从二维空间可以得到三个气泡轮廓。每个气泡的稳定性由线性化算子的特征值确定。利用稳定的气泡轮廓，当参数选择适当时，我们可以在一般有界域上构造一个由扰动球组成的静止集合。
The FitzHugh--Nagumo system gives rise to a nonlocal geometric variational problem defined on subsets of a domain. The energy of a subset contains three terms: its perimeter, its volume, and a long-range self-interaction term represented by the integral of the solution to a screened Poisson's equation. A bubble profile is a ball-shaped stationary set when the domain is the entire space. If the space dimension is three or higher, depending on the parameters of the problem, there can be zero, one, or two bubble profiles. This is in contrast to an earlier result for the two-dimensional space, from which one may have three bubble profiles. The stability of each bubble is determined from the eigenvalues of the linearized operator. Using a stable bubble profile, one constructs a stationary assembly of perturbed balls on a general bounded domain, when the parameters are properly chosen.