论文标题
Gierer-Meinhardt模型的局部3-D点模式的渐近分析：存在，线性稳定性和缓慢的动力学
An Asymptotic Analysis of Localized 3-D Spot Patterns for Gierer-Meinhardt Model: Existence, Linear Stability and Slow Dynamics
论文作者
论文摘要
局部斑点模式，其中一个或多个溶液组分集中在域中的某些点上，是反应扩散系统的一类局部模式，它们在广泛的建模场景中出现。在任意界限的3-D结构域中，分析了众所周知的奇异扰动的Gierer-Meinhardt（GM）激活剂抑制剂系统的局部多点模式的存在，线性稳定性和缓慢的动力学，在小型激活剂扩散剂扩散剂$ \ varepsilon^2 \ ll ll 1 $ 1 $ 1 $中。我们的主要重点是针对抑制剂扩散性的不同渐近范围$ d $对不同类型的多点模式进行分类，并预测其线性稳定性。对于范围，$ d = {\ mathcal o}（\ varepsilon^{ - 1}）\ gg 1 $，尽管可以构建对称和不对称的准e平衡点模式，但不对称模式被证明总是不稳定的。在$ d $的这一范围内，可以表明对称点模式可以进行竞争不稳定性或HOPF分叉，从而导致斑点歼灭或时间斑点振幅振荡。对于$ d = {\ mathcal o}（1）$，仅存在对称点quasi equilibria，并且它们在$ {\ Mathcal o}（1）$时间间隔上是线性稳定的。在此范围内，结果表明，斑点位置在$ {\ Mathcal O}（\ Varepsilon^{ - 3}）$时尺度在其平衡位置的$ {\ Mathcal O}（\ varepsilon^{ - 3}）上缓慢演变，该$ time time scale朝向其平衡位置，该梯度流由涉及减少波绿色功能的离散能量确定。特定核心问题的远场行为的核心作用是特征局部位点的特征，用于在$ d = {\ Mathcal O}（1）$和$ d = {\ Mathcal O}（\ varepsilon^{ - 1}）$ semimes的属性属性的属性中，在$ d = {\ Mathcal o}（1）$和$ d = {\ Mathcal o}（\ Mathcal O}（\ Mathcal O}（\ Mathcal O}（\ Mathcal O}（\ Mathcal O}）中
Localized spot patterns, where one or more solution components concentrates 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. In an arbitrary bounded 3-D domain, the existence, linear stability, and slow dynamics of localized multi-spot patterns is analyzed for the well-known singularly perturbed Gierer-Meinhardt (GM) 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 multi-spot 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 towards 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.
