2003 Fiscal Year Final Research Report Summary
Singular Integral Operators and Nonlinear Problems
| Project/Area Number |
13440041
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | University of Tsukuba |
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Principal Investigator |
TAIRA Kazuaki University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (90016163)
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| Co-Investigator(Kenkyū-buntansha) |
MIYACHI Akihiko Tokyo Women's University, Professor, 文理学部, 教授 (60107696)
KINOSHITA Tamotu University of Tsukuba, Institute of Mathematics, Assistant Professor, 数学系, 講師 (90301077)
WAKABAYASHI Seiichiro University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (10015894)
NAKAMURA Gen Hokkaido University, Faculty of Science, Professor, 理学部, 教授 (50118535)
YAGI Atsushi Osaka University, Faculty of Science, Professor, 工学部, 教授 (70116119)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Singular Integral Operator / Nonlinear Boundary Value Problem / Population Dynamics / Combustion Problem / Feller Semigroup / Markov Process |
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| Research Abstract |
Our results may be summarized as follows
1.First, we studied from the viewpoint of functional analysis the problem of construction of Markov processes with boundary conditions in probability theory. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of singulars integral operators. We constructed a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the set where the particle is definitely absorbed
2.Secondly, we studied existence and uniqueness problems of positive solutions of diffusive logistic equations with indefinite weights which model population dynamics in environments with strong spatial heterogeneity. We proved drat the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. Moreover, we discussed the stability properties for positive steady states
3.Thirdly, we studied semilinear elliptic boundary value problems arising in combustion theory which obey the supple Arrhenius rate law and Newtonian cooling. We proved that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless heat evolution rate
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