1999 Fiscal Year Final Research Report Summary
Theory and Applications of Evolution Equations with Nonlocal Conditions
| Project/Area Number |
10640172
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Shimane University |
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Principal Investigator |
KATO Nobuyuki Dept. of Math., Shimane University, Associate Professor, 総合理工学部, 助教授 (40177423)
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| Co-Investigator(Kenkyū-buntansha) |
SUGIE Jitsuro Dept. of Math., Shimane University, Professor, 総合理工学部, 教授 (40196720)
AIKAWA Hiroaki Dept. of Math., Shimane University, Professor, 総合理工学部, 教授 (20137889)
FURUMOCHI Tetsuo Dept. of Math., Shimane University, Professor, 総合理工学部, 教授 (40039128)
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| Project Period (FY) |
1998 – 1999
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| Keywords | nonlocal / evolution equation / muscle contraction / population dynamics / size-dependent |
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| Research Abstract |
We have investigated two different types of evolution equations, focusing on the common key word "nonlocal conditions".
[Muscle contraction models]
The muscle contraction is a consequence of relative sliding between the thick filament (Myosin) and the thin filament (Actin).
This sliding occurs when the so-called cross-bridges attach Myosin to Actins, and act as spring.
The Muscle contraction model describes the temporal variation of the density of the attached cross-bridges.
Since the contraction speed is dependent on the number of the attached cross-bridges, a nonlocal term appears in the model.
In this research, we have considered transport equations of hyperbolic type and transport-diffusion equations of parabolic type.
We obtained the existence and uniqueness results with some general setting allowing the attaching and detaching of the cross-bridges being nonlinear.
[Population dynamics]
The population model depending on the individuals size and time has been investigated as a natural model describing the population of plants in forests or plantations.
The birth law includes a nonlocal term.
We obtained some results concerning the existence of positive solutions and blow up and global existence of the solution to a general model.
I had a talk on these results in the international conference held at Newport Beach, USA in 1998.
Also, we have shown the continuous dependence on the birth and aging functions as well as initial data.
Besides, we have investigated the model having the nonlinear growth rate and obtained the existence result.
I had a talk on this in the international conference held at Berlin, Germany in 1999.
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