Research of structure of stable spike solutions of Gierer-Meinhardt system in non-oonvex domain in two space dimensions

本文从反应扩散系统模式形成问题的角度研究了Gierer-Meinhardt系统稳定尖峰解的结构。我们对列色刚现象的模式形成特别感兴趣，这是1896年由德国列色刚教授发现的。我们首先对Keller-Rubinow模型进行了模拟，验证了时间定律和空间定律的成立，并利用模型方程对时间定律和空间定律进行了严格的数学证明。其次，我们注意到Keller-Rubinow模型不能实现降水的宽度。因此，我们利用九州大学S.Kai教授提出的胶体生长和溶解理论对模型方程进行了改进。这是一个很好的模型，因为它实现了真实的化学实验。特别是实现了环形图案的分裂和破坏现象。最后，我们推测该系统具有本质的不稳定性，它会导致环形图案的分裂和破坏现象，并且最终的图案应该是类似棋盘的图案。这是一个非常有趣的结果，所以我们在一些国际会议和研讨会上讨论了它，并在参考文献中报道了它。

We have studied about structure of stable spike solutions of Gierer-Meinhardt system from the view point of pattern formation problem in reaction-diffusion system. We have been especially interested in pattern formation of Liesegang phenomena, which was discovered in 1896 by Professor Liesegang in Germany. We first made simulations of Keller-Rubinow model to verify that time law and spacing law hold and we made a mathematically rigorous proof of them by use of the model equation. Next, we noticed that Keller-Rubinow model cannot realize width of precipitation. Therefore, we improved the model equation by use of the theory of colloid growth and dissolution progressed by Professor S.Kai in Kyushu University. This is a very good model because it realizes the real chemical experiment. Especially, it realize the splitting and destroying phenomena of the ring pattern. Finally, we make a conjecture that the system has an essential instability and it causes the splitting and destroying phenomena of the ring pattern, and moreover the final pattern should be the checker board like pattern. This is a very interested result, so we talk about it in some international conferences and symposiums to report it in references in the below 11..