Internal Transition Layrs for semilinear elliptic systems and a related free boundary problem
Project/Area Number  09640194 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
解析学

Research Institution  HIROSHIMA UNIVERSITY 
Principal Investigator 
SAKAMOTO Kunimochi Hiroshima University, Faculty of Science, Associate Professor, 理学部, 助教授 (40243547)

Project Period (FY) 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,200,000 (Direct Cost : ¥2,200,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1997 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  Internal Transition Layrs / Interface / Free Boundary Problem / Asymptotic Expansion / ReactionDiffusion System / Singular Perturbation / 特異摂動 / 界面方程式 / 高次元界面 / 特異摂動法 
Research Abstract 
In this research project, the study on the existence of stationary internal layer solutions for singularly perturbed systems of reactiondiffusion 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 freeboundary 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 reactiondiffusion system. At the same time, a sufficient condition for a solution gamma of the free boundary problem to be an interface of internal tran
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sition layer solutions was characterized as the invertibility of a nonlocal 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

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Research Products
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