In this paper, we study the existence and stability of multiple spikes pattern to the fractional Gierer–Meinhardt model with periodic boundary conditions and the fractional power s = 1 2. Specifically, we rigorously establish the existence of symmetric multiple spikes and asymmetric two-spikes solutions by the classical Lyapunov–Schmidt reduction method. We also investigate the stability of the constructed solution by studying its associated large and small eigenvalue problems, where we need to consider two nonlocal eigenvalue problems in their fractional versions. In the study of the large eigenvalue problem, the quantity DK(𝜖) = 2 πKlog1 𝜖 is the critical threshold which determines the stability of K-peaked solutions. For the symmetric two-spikes pattern we obtain the asymptotic expansion for the critical threshold DK(𝜖) up to the second order. Moreover, we provide some elementary properties of the Green’s function, including the first and second derivatives, they are linked to the location of the spikes and the stability. Among these properties on the Green’s function, we find out that the polygamma function ϕ(x) = d dxlogΓ(x) plays a crucial role.

中文翻译：

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一维域中半拉普拉斯Gierer-Meinhardt系统对称和非对称模式的存在性和稳定性

在本文中，我们研究了具有周期性边界条件和分数功率的分数 Gierer-Meinhardt 模型的多个尖峰模式的存在和稳定性s = 1 2. 具体来说，我们通过经典的 Lyapunov-Schmidt 归约方法严格建立了对称多尖峰和不对称双尖峰解的存在。我们还通过研究其相关的大小特征值问题来研究构造解决方案的稳定性，其中我们需要考虑两个非局部特征值问题的分数版本。在研究大特征值问题时，量Dķ(𝜖) = 2 πķ日志1 𝜖是决定稳定性的临界阈值ķ-峰值解决方案。对于对称的双尖峰模式，我们获得了临界阈值的渐近展开Dķ(𝜖)直到第二个订单。此外，我们提供了格林函数的一些基本性质，包括一阶和二阶导数，它们与尖峰的位置和稳定性有关。在格林函数的这些性质中，我们发现多伽马函数φ(X) = d dX日志Γ(X)起着至关重要的作用。