On the existence and stability of singularly perturbed solutions for one-activator and two-inhibitors reaction-diffusion models

这个项目的目的是找到一个三组分反应扩散系统的定常解，并用奇异摄动方法检验解的稳定性。在这些过程中，最困难的问题是求解一个约化问题，即求一对具有间断非线性的椭圆型边值问题的C^1‘类解，并知道解对物理参数的依赖性。从动力系统的观点出发，将约化问题转化为具有不连续非线性的四维动力系统。在R^4中存在两个双曲平衡点，它们都有二维稳定流形和二维不稳定流形。上述问题对应于在受限的二维空间R^2上找到一个连续轨迹。在人工条件下，我们成功地找到了这样的轨迹。并在此基础上，利用奇异摄动方法证明了原问题解的存在性。接下来，我们在二维区域R^2上考虑同样的问题。在类似于上面的条件下，我们可以找到一个径向对称解。最后，我们研究了围绕解的线性化特征值问题，并通过数值模拟跟踪了临界特征值。我们检查了不稳定模的出现，即第0模和第一模。前者对应于行波解(即行波点)从驻波解的分叉现象，后者对应于非对称解从径向对称驻波解的分叉现象。但我们找不到发生Hopf分叉的参数。

Our purpose of this project is to find a stationary solution of a three-component reaction-diffusion system and examine the stability of the solution by applying a singular perturbation method. The most difficult problem in these process is to solve a reduced problem, that is, to find a C^1'-class solution of the pair of elliptic boundary value problems with discontinuous nonlinearities and to know the dependency of the solution on physical parameters. With a dynamical system view point, the reduced problem is rewritten as a four-dimensional dynamical system with discontinuous nonlinearities. There are two hyperbolic equilibrium points, both have two-dimensional stable manifolds and two-dimensional unstable manifolds, in R^4. The above problem corresponds to find a continuous trajectroy on the restricted two-dimensional space R^2. Under an artificial condition, we succeed in finding such trajectroy. And using this, we show the existence of a solution of the original problem applying a singular perturbation method. Next we consider the same problem on the two-dimensional region R^2. Under a similar condition to the above, we can find a radialy symmetric solution. Finally we study the linearized eigenvalue problem around the solution and trace the, critical eigenvalue by numerical simulation. We check the appearance of instability modes, that is, the 0-th mode and the first mode. The former case corresponds to the bifurcation phenomena of a travelling wave solution, that is, a traveling spot, from a standing wave solution and the latter case does to the bifurcation of an asymmetric solution from a radialy symmetric standing wave solution. But we cannot find a parameter on which the Hopf bifurcation occurs.