2004 Fiscal Year Final Research Report Summary
Qualitative theory and asymptotic analysis of nonlinear partial differential equations
| Project/Area Number |
13440028
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MATANO Hiroshi The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40126165)
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| Co-Investigator(Kenkyū-buntansha) |
FUNAKI Tadahisa The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (60112174)
YAMAMOTO Masahiro The University of Tokyo, Graduate School of Mathematical Sciences, Associate professor, 大学院・数理科学研究科, 助教授 (50182647)
WEISS Georg The University of Tokyo, Graduate School of Mathematical Sciences, Associate professor, 大学院・数理科学研究科, 助教授 (30282817)
EI Shin-Ichiro Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (30201362)
TANIGUCHI Masaharu Tokyo Institute of Technology, Graduate School of Information Science and Technology, Associate professor, 大学院・情報理工学研究科, 助教授 (30260623)
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| Project Period (FY) |
2001 – 2004
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| Keywords | nonlinear partial differential equation / qualitative theory / asymptotic analysis / blow-up of solutions / singularity / singular limit / traveling wave / inverse problem |
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| Research Abstract |
We have considered the behavior of solutions after the blow-up time for nonlinear heat equations with a power nonlinearity and those with an exponential nonlinearity. It is shown that singularities that apper at the blow-up time disappear and the solutions become smooth immediately (Matano, SIAM J.Math.Anal., in press). We have also studied the blow-up rate for nonlinear heat equations with a power nonlinearity and proved that the blow-up is always type 2 so far as the power is in the intermediate supercritical range (Matano, Comm.Pure Appl.Math., 2004).
Yamamoto has studied an inverse problem of determining two unknown convection terms in a two-dimensional elliptic equation, and proved that those terms can be determined by the so-called Dirichlet-Neumann map (Inverse Problems, 2004).
Weiss has considered a singular limit problem for parabolic equations that are applicable to the double obstacle problem. Using a new monotonicity formula, he has succeeded in exstimating the Hausdorff dimension of the free boundary (Calc.Var.PDE, 2003).
Ei has studied a reaction-diffusion system on a two-dimensional cylinder and analysed the behavior of solutions having a pulse-like profile. He derived an equation governing the motion of slow pulses and proved that traveling pulses are mutually repelling (DCDS, Ser.A, in press).
Taniguchi has considered the so-called "singular limit eigenvalue problem method" (SLEP method), which is a powerful tool in analyzing the stability of stationary solution of the singular limit problem. He has generalized this method so that it applies to problems in unbounded domains. Using this result, he has proved the stability of planar traveling waves in a bistable reaction-diffusion system (DCDS, Ser.B, 2003).
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