论文标题
1D影子Gierer-Meinhardt问题的单尖峰解决方案
Single-spike solutions to the 1D shadow Gierer-Meinhardt problem
论文作者
论文摘要
表现出图灵类型模式形成的反应扩散系统的一个基本示例是Gierer-Meinhardt系统，该系统在适当的奇异极限中降低了阴影Gierer-Meinhardt问题。由于其在各种生物学应用中的适用性，在过去的几十年中，通过严格，渐近和数值方法对这种奇异的扰动问题进行了广泛研究。但是，标准匹配的渐近方法不适用于（NI 1998，WEI 1998），因此通常缺乏单峰溶液的分析表达式。 通过基于广义双曲线功能引入ANSATZ，我们确定了任何$ 1 <p <\ infty $的一维阴影Gierer-Meinhardt问题的精确径向对称溶液，这取决于内部和边界尖峰解决方案，具体取决于峰位置。我们的方法不仅证实了文献中存在的数值结果，而且还提供了基于不同边界条件（例如混合）和/或$ n $维域的不同边界条件（例如混合）和/或$ n $维域来解决阴影Gierer-Meinhardt问题的指导。
A fundamental example of reaction-diffusion system exhibiting Turing type pattern formation is the Gierer-Meinhardt system, which reduces to the shadow Gierer-Meinhardt problem in a suitable singular limit. Thanks to its applicability in a large range of biological applications, this singularly perturbed problem has been widely studied in the last few decades via rigorous, asymptotic, and numerical methods. However, standard matched asymptotics methods do not apply (Ni 1998, Wei 1998), and therefore analytical expressions for single spike solutions are generally lacking. By introducing an ansatz based on generalized hyperbolic functions, we determine exact radially symmetric solutions to the one-dimensional shadow Gierer-Meinhardt problem for any $1 < p < \infty$, representing both inner and boundary spike solutions depending on the location of the peak. Our approach not only confirms numerical results existing in literature, but also provides guidance for tackling extensions of the shadow Gierer-Meinhardt problem based on different boundary conditions (e.g. mixed) and/or $n$-dimensional domains.
