Internal Transition Layrs for semilinear elliptic systems and a related free boundary problem

在本研究项目中，研究了高维奇摄动反应扩散方程组的定常内层解的存在性。在假定自由边界问题存在解Gamma的情况下，证明了存在一族内层解，其过渡界面恰好就是解的Gamma。在证明过程中，还建立了在高维域中渐近展开内部过渡层的一般方法。在此基础上，导出了定义在界面上的椭圆型方程组，并证明了该椭圆型方程组的可解性等价于原反应扩散系统内过渡层的存在。同时，给出了自由边界问题的解Gamma是内部变换…界面的一个充分条件多层解被刻画为定义在Gamma上的非局部一阶椭圆算子的可逆性。将上述一般结果应用于区域具有高度对称性的情形，如球和环空，从而证明了内部过渡层的存在及其不稳定性。为了使上述一般理论有效，有一个关键的假设，即由问题的非线性确定的某个量J‘是正的。当这个量J‘为负时，还研究了当奇异摄动参数趋于零时，存在无穷多个静态分歧点的结论。这也表明，当J‘&lt；0时，单一极限系统是无限退化的。对于一般有界域，自由边界问题的解的存在性远不是完全的，而且由于问题的非局部性质，它甚至不能很好地用昂贵的数学术语来表示。这些都是我们未来研究项目的目标。较少

In this research project, the study on the existence of stationary internal layer solutions for singularly perturbed systems of reaction-diffusion equations in high dimensional domains was carried out. The summary of the results obtained is as follows.Assuming the existence of a solution gamma for a free-boundary problem, it was shown that there exists a family of internal layer solutions whose transition interface is precisely the solution gamma. In the process of the proof, also established was a general method to asymptotically expand internal transition layers in, high dimensional domains. Moreover, a system of elliptic equations defined on the interface gamma was derived, and then it was shown that the solvability of this system of elliptic equation is equivalent to the existence of the internal transition layers for the original reaction-diffusion system. At the same time, a sufficient condition for a solution gamma of the free boundary problem to be an interface of internal tran … More sition layer solutions was characterized as the invertibility of a non-local first order elliptic operator defined on gamma. The general results above was applied to the situation where the domain has a high degree of symmetry, such as balls and annuli, giving rise to establish the existence of internal transition layers and their instability.In order for the general theory above to be valid, there was a crucial assumption that a certain quantity J' determined by the nolinearity of the problem be positive. When this quantity J' is negative, an investigation was also made that indicates the existence of infinitely many static bifurcation points as the singular perturbation parameter tends to zero. This also suggests that when J' <0 the singlar limit system is infinitely degenerated. For general bounded domain, the existence of the solution to the free boundary problem is far from being complete, and owing to the nonlocal nature of the problem it is not even well formulated in pricese mathematical terms. These are our targets of future research projects. Less