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

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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