论文标题
具有非零激活因子磁通的Gierer-Meinhardt系统中的多峰值图案
Multi-Spike Patterns in the Gierer-Meinhardt System with a Non-Zero Activator Boundary Flux
论文作者
论文摘要
在过去的几十年中，对奇异扰动的反应扩散方程的局部溶液的结构，线性稳定性和动力学一直是许多严格，渐近和数值研究的重点。但是，除了少数例外，这些研究经常假定均匀的边界条件。由于最近关注散装耦合问题的分析，我们考虑了不均匀的诺伊曼边界条件对激活因子在奇异扰动的一维的Gierer-Meinhardt反应系统中的影响。我们表明，这些边界条件需要形成集中在域边界附近边界层中的尖峰。使用匹配的渐近扩展方法，我们构建边界层尖峰并得出了一类新的移动非局部特征值问题，我们严格地证明了部分稳定性结果。此外，通过使用渐近，严格和数值方法的组合，我们研究了选定的单尖峰和两种尖峰模式的结构和线性稳定性。特别是，我们发现不均匀的诺伊曼边界条件增加了存在不对称的两尖峰模式并且稳定的参数值范围。
The structure, linear stability, and dynamics of localized solutions to singularly perturbed reaction-diffusion equations has been the focus of numerous rigorous, asymptotic, and numerical studies in the last few decades. However, with a few exceptions, these studies have often assumed homogeneous boundary conditions. Motivated by the recent focus on the analysis of bulk-surface coupled problems we consider the effect of inhomogeneous Neumann boundary conditions for the activator in the singularly perturbed one-dimensional Gierer-Meinhardt reaction-diffusion system. We show that these boundary conditions necessitate the formation of spikes that concentrate in a boundary layer near the domain boundaries. Using the method of matched asymptotic expansions we construct boundary layer spikes and derive a new class of shifted Nonlocal Eigenvalue Problems for which we rigorously prove partial stability results. Moreover by using a combination of asymptotic, rigorous, and numerical methods we investigate the structure and linear stability of selected one- and two-spike patterns. In particular we find that inhomogeneous Neumann boundary conditions increase both the range of parameter values over which asymmetric two-spike patterns exist and are stable.
