2013 Fiscal Year Final Research Report
A study on the elliptic and the parabolic equations associated with noncompact variational structures
| Project/Area Number |
23540232
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Fukushima University |
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Principal Investigator |
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| Project Period (FY) |
2011 – 2013
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| Keywords | 非線型解析 / 変分法 / 楕円型方程式 / 放物型方程式 / エネルギー汎関数 / リャプノフ汎関数 / 非コンパクト性 |
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| Research Abstract |
In this research, we are concerned with the asymptotic behavior of time-global solutions for semilinear parabplic equations whose Lyapnov functional are suffered from the lack of compactness, togehter with the elliptic problem with variational functional with lack of compactness. Also we studeid the variational problem (minimizing problem) associated with the critical functional inequalities such as Sobolev, Hardy and the Trudinger-Moser type.
As for the parabolic problems, we treated the semilinear parabolic problem involving critical Sobolev exponent and showed the energy quantization phenomena for the time-global solutions. Also we studied the elliptic problem defined in the exterior domain with bounded complement with weight function decaying at the spatial infinity. We proved that if the decay of the weight function is sufiiciently slow, then the equations have multiple positive solutions.
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