SIAM Journal on Applied Mathematics, Volume 81, Issue 2, Page 378-406, January 2021.
Localized spot patterns, where one or more solution components concentrate at certain points in the domain, are a common class of localized pattern for reaction-diffusion systems, and they arise in a wide range of modeling scenarios. Although there is a rather well-developed theoretical understanding for this class of localized pattern in one and two space dimensions, a theoretical study of such patterns in a three-dimensional setting is, largely, a new frontier. In an arbitrary bounded three-dimensional domain, the existence, linear stability, and slow dynamics of localized multispot patterns are analyzed for the well-known singularly perturbed Gierer--Meinhardt activator-inhibitor system in the limit of a small activator diffusivity $\varepsilon^2\ll 1$. Our main focus is to classify the different types of multispot patterns and predict their linear stability properties for different asymptotic ranges of the inhibitor diffusivity $D$. For the range $D={\mathcal O}(\varepsilon^{-1})\gg 1$, although both symmetric and asymmetric quasi-equilibrium spot patterns can be constructed, the asymmetric patterns are shown to be always unstable. On this range of $D$, it is shown that symmetric spot patterns can undergo either competition instabilities or a Hopf bifurcation, leading to spot annihilation or temporal spot amplitude oscillations, respectively. For $D={\mathcal O}(1)$, only symmetric spot quasi-equilibria exist and they are linearly stable on ${\mathcal O}(1)$ time intervals. On this range, it is shown that the spot locations evolve slowly on an ${\mathcal O}(\varepsilon^{-3})$ time scale toward their equilibrium locations according to an ODE gradient flow, which is determined by a discrete energy involving the reduced-wave Green's function. The central role of the far-field behavior of a certain core problem, which characterizes the profile of a localized spot, for the construction of quasi-equilibria in the $D={\mathcal O}(1)$ and $D={\mathcal O}(\varepsilon^{-1})$ regimes, and in establishing some of their linear stability properties, is emphasized. Finally, for the range $D={\mathcal O}(\varepsilon^{2})$, it is shown that spot quasi-equilibria can undergo a peanut-splitting instability, which leads to a cascade of spot self-replication events. Predictions of the linear stability theory are all illustrated with full PDE numerical simulations of the Gierer--Meinhardt model.

中文翻译：

---

Gierer-Meinhardt模型的局部三维光斑图案的渐近分析：存在性，线性稳定性和慢动力学

SIAM应用数学杂志，第81卷，第2期，第378-406页，2021年1月。
在反应扩散系统中，一种或多种溶液组分集中在域中某些点的局部斑点模式是一类常见的局部模式，它们出现在各种各样的建模场景中。尽管对于此类在一维和二维空间中的局部模式有相当完善的理论理解，但在三维环境中对此类模式进行理论研究在很大程度上是一个新的前沿领域。在任意有界的三维域中，分析了已知的奇摄动的Gierer-Meinhardt激活剂-抑制剂系统在小激活剂扩散率$ \ varepsilon的限制下局部多点模式的存在，线性稳定性和慢动力学^ 2 \ ll 1 $。我们的主要重点是对不同类型的多点模式进行分类，并针对抑制剂扩散率$ D $的不同渐近范围预测其线性稳定性。对于$ D = {\ mathcal O}（\ varepsilon ^ {-1}）\ gg 1 $范围，尽管可以构造对称和非对称的准平衡光斑图样，但非对称图样始终显示为不稳定。在$ D $的这一范围内，表明对称光斑图案可能经历竞争不稳定性或Hopf分叉，分别导致光斑spot灭或时间光斑振幅振荡。对于$ D = {\ mathcal O}（1）$，仅存在对称点拟均衡，并且它们在$ {\ mathcal O}（1）$时间间隔内线性稳定。在这个范围内 结果表明，光点位置根据ODE梯度流在$ {\ mathcal O}（\ varepsilon ^ {-3}）$时间尺度上朝其平衡位置缓慢演化，该梯度流由包含-wave Green的功能。某个核心问题的远场行为的中心作用，它表征了局部斑点的轮廓，对于$ D = {\ mathcal O}（1）$和$ D = {强调了\ mathcal O}（\ varepsilon ^ {-1}）$体制，并建立了它们的某些线性稳定性。最后，对于$ D = {\ mathcal O}（\ varepsilon ^ {2}）$范围，表明准准平衡点可能经历花生分裂不稳定性，从而导致级联的现场自我复制事件。