2007 Fiscal Year Final Research Report Summary
The relation between the solutions and the nonlinearities in reaction diffusion systems
| Project/Area Number |
18540147
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Ryukoku University |
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Principal Investigator |
NINOMIYA Hirokazu Ryukoku University, Department of Applied. Mathematics and Informatics, Associate Professor (90251610)
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| Co-Investigator(Kenkyū-buntansha) |
IIDA Masato Iwate University, Department of Mathematics, Associate Professor (00242264)
MORITA Yoshihisa Ryukoku University, Department of Applied Mathematics and Informatics, Full Professor (10192783)
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| Project Period (FY) |
2006 – 2007
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| Keywords | reaction-diffusion system / traveling wave / entire solution / blowup |
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| Research Abstract |
We studied the following three topics :
1. Characterization of entire solutions of Allen-Cahn equations
A solution which exists for all time is called by entire solution. Since the parabolic system is well-posed only for positive time in a suitable functional space in general, the entire solution is contained in the attraction I and one of the investigators, Professor Morita, succeeded in constructing the new type of entire solutions. This entire solution behaves two traveling wave solution as t tends to-infinity. The entire solution in the Allen-Cahn equation with the balanced case is also constructed with Professors Guo and Chen. It is shown with Professors Chen, Guo Hamel and Roqujoffre that the existence of the traveling wave solution in the balanced Allen-Cahn equation in the higher dimensional space.
2. Nonlinear diffusion and reaction-diffusion approximation
It is shown with Professors lids and Kimura that the cross-diffusion system is approximated by some reaction-diffusion system by splitting the species into the fast stet and the slow one. Using this approximation we revealed the relationship between Turing's instability and the cross-diffusion instability.
3. Characterization of nonlinearities in the blowup problems of reaction-diffusion systems
It is called a linearity-induced blowup that the system with linear perturbation possesses a solution which blows up in a finite time, while all solutions of the original system exist for any positive time. The example of linearity-induced blowup was constructed with Professor Weinberger even if the linear perturbation is restricted to the purely inward one. We also studied the sufficient condition far linearity-induced blowup for p-homogeneous nonlinearity.
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