Study on nonlinear partial differential equations with set-valued perturbations
| Project/Area Number |
15K13451
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Mathematical analysis
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| Research Institution | Waseda University |
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Principal Investigator |
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| Research Collaborator |
AKAGI GOROU 東北大学, 理学研究科, 教授 (60360202)
ISHIWATA MICHINORI 大阪大学, 基礎工学研究科, 教授 (30350458)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 集合値関数 / 非線形偏微分方程式 / 劣微分作用素 / 非線形発展方程式 / 非線形放物型方程式 / 非線形楕円型方程式 |
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| Outline of Final Research Achievements |
We studied the existence of solutions of the initial value problem and the time periodic problem for the following semilinear parabolic equations: du/dt - △u + β(u) + G(x,t,u) = f(x,t) with the homogeneous Dirichlet boundary condition. Here β(u) is a (multi-valued) monotone operator and the perturbation term G(x,t,u) is a upper semi-continuous or lower semi-continuous set-valued function, which is a generalized notion of continuous function. When G is a set-valued function, there was no results even for super-linear cases. In our study, we succeeded in generalize the best result for the case where G is single-valued to the case where G is a set-valued function.
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| Academic Significance and Societal Importance of the Research Achievements |
多価写像に関する解析学は，位相数学，測度論，非線形関数解析，応用数学などのいろいろな数学分野が交錯する興味深い分野の一つである．集合値写像の数学的重要性は，20 世紀初頭に既に認識されはじめ， Hausdorff, Vietoris, Hahn, Kuratowski などの多くの数学者によって研究されてきた．これ等の成果は，常微分方程式論(ODE)に引き継がれ，1960 年代から，精力的に研究が進められ，膨大な知見が蓄積されてきた．一方で偏微分方程式論に於いては，殆ど研究されてはこなかった．本研究は，ODEで蓄積されてきた知見を偏微分方程式論に移植する本格的な試みであり，その意義は大きい．
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Report
(5 results)
Research Products
(34 results)
