2006 Fiscal Year Final Research Report Summary
On the singularity of solutions for nonlinear parabolic equation
| Project/Area Number |
17540189
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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 |
Global analysis
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| Research Institution | Tokyo Gakugei University |
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Principal Investigator |
MIZOGUCHI Noriko Tokyo Gakugei University, Education, Associate Prof., 教育学部, 助教授 (00251570)
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| Co-Investigator(Kenkyū-buntansha) |
YANGIDA Eiji Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80174548)
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| Project Period (FY) |
2005 – 2006
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| Keywords | semilinear heat equation / incomplete blowup / supercritical exponent / multiple blowup / type II blowup / blowup rate |
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| Research Abstract |
The behavior of solutions to a semilinear heat equation with power nonlinearity dramatically changes before and after the Sobolev critical exponent. They are investigated in detail in the subcritical case, but there are only a few results in the supercritical case. Incomplete blowup, growup of a global solution and type II blowup are typical phenomena in the superciritical case.
The existence of a incomplete blowup solution is known, but the behavior after blowup has reminded open. We obtained a weak solution which blowup twice in the classical sense, and extended it to general multiple blowup. These solutions blow up at the same point (at the origin) at each blowup time and the difference between two successive blowup times is sufficiently large. We studied the existence of a solution which blows up at different places at different blowup times for given blowup times.
By Galaktionov-Vazquez and myself, there were peaking solution which converges to 0 as time tens to infinity. We gave peaking solutions with various behaviors at time infinity.
We constructed growup solutions with exact growup rate applying a method based on infinite-dimensional dynamical system. It also gave the optimal growup rate of solutions below the singular steady state.
Herrero-Velazquez showed that there exists a solution which undergoes type II blowup. However their proof is very long and complicated. We got a type II blowup solution as a separatirix between global solutions and blowup solutions. Our proof is simpler than the previous one.
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Research Products
(29 results)
