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Affiliation |
Faculty of Education Mathematics education |
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Title |
Associate Professor |
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External Link |
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HIRAYAMA Hiroyuki
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Research Areas 【 display / non-display 】
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Natural Science / Mathematical analysis / Partial differential equations
Papers 【 display / non-display 】
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Hiroyuki Hirayama and Yasuyuki Oka
Journal of Differential Equations 412 214 - 249 2024.12
Publishing type：Research paper (scientific journal) Publisher：Journal of Differential Equations
The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups G with the homogeneous dimension N. We consider the nonlinear function behaves like |u|α or |u|α−1u (α>1) and the initial data u0 belongs to the Sobolev spaces Lsp(G) for 1<p<∞ and 0<s<N/p. Since stratified Lie groups G include the Euclidean space Rn as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on Rn to G. It should be noted that our proof is very different from it given by Ribaud on Rn. We adopt the generalized fractional chain rule on G to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on Rn. By using the generalized fractional chain rule on G, we can avoid the discussion of Fourier analysis on G and make the proof more simple.
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Ikki Fukuda, Hiroyuki Hirayama
Nonlinear Analysis: Real World Applications 79 2024.10
Publishing type：Research paper (scientific journal) Publisher：Nonlinear Analysis: Real World Applications
We consider the Cauchy problem for the generalized Zakharov–Kuznetsov–Burgers equation in 2D. This is one of the nonlinear dispersive–dissipative equations, which has a spatial anisotropic dissipative term −μuxx. In this paper, we prove that the solution to this problem decays at the rate of [Formula presented] in the L∞-sense, provided that the initial data u0(x,y) satisfies u0∈L1(R2) and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the L∞-norm of the solution. As a result, we prove that the given decay rate [Formula presented] of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schrödinger equation, we derive the explicit asymptotic profile for the solution.
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Hiroyuki Hirayama, Masahiro Ikeda
Calculus of Variations and Partial Differential Equations 63 ( 7 ) 2024.9
Publishing type：Research paper (scientific journal) Publisher：Calculus of Variations and Partial Differential Equations
We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension.
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Sharp well-posedness for the Cauchy problem of the two dimensional quadratic nonlinear Schrödinger equation with angular regularity Reviewed
Hiroyuki Hirayama, Shinya Kinoshita, Mamoru Okamoto
Journal of Differential Equations 234 2024.2
Language：English Publishing type：Research paper (scientific journal)
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Existence and uniqueness for the Cauchy problem of semilinear heat equations on stratified Lie groups in the critical Sobolev space Reviewed
Hiroyuki Hirayama and Yasuyuki Oka
Taiwanese J. Math. Advance Publication 1 - 21 2024
Publishing type：Research paper (scientific journal)
DOI： 10.11650/tjm/240604
Books 【 display / non-display 】
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工科系のための偏微分方程式入門
岡 康之, 平山 浩之, 鈴木 俊夫, 藤ノ木 健介（ Role： Joint author）
学術図書出版社 2023.3
Book type：Scholarly book
MISC 【 display / non-display 】
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微分型非線形シュレディンガー方程式系のほとんど最良なソボレフ空間における適切性について Invited
平山 浩之
第60回 実函数論・函数解析学合同シンポジウム 講演集 34 - 53 2021.9
Language：Japanese
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Well-posedness for a system of quadratic derivative nonlinear Schr¨odinger equations with periodic initial data at the scaling critical regularity
平山 浩之
第5 回白浜研究集会報告集 99 - 108 2014.2
Language：Japanese
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トーラス上の高階次分散型方程式の時間局所適切性について
平山 浩之
第31 回発展方程式若手セミナー 報告集 223 - 236 2009.12
Language：Japanese
Presentations 【 display / non-display 】
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Variational problems for the system of nonlinear Schrodinger equations with quadratic derivative nonlinearities Invited
平山 浩之
Takamatsu Workshop on Differential Equations and Related Topics
Event date： 2024.3.26 - 2024.3.27
Presentation type：Oral presentation (general)
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一般化 Zakharov–Kuznetsov–Burgers 方程式の初期値問題の解の長時間挙動と最良な減衰評価について
福田 一貴, 平山 浩之
日本数学会2023年度秋季総合分科会 (中央大学) 2023.3.17
Event date： 2023.9.20 - 2023.9.23
Presentation type：Oral presentation (general)
Venue：中央大学
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Existence and stability of the ground states to the system of nonlinear Schrodinger equations with derivative nonlinearity Invited
平山 浩之
応用解析研究会
Event date： 2023.5.27
Presentation type：Oral presentation (general)
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Large time behavior and optimal decay estimate for solutions to the generalized KP-Burgers equation Invited
平山 浩之
NLPDEセミナー
Event date： 2023.5.26
Presentation type：Oral presentation (general)
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Existence and stability of the ground states to the system of nonlinear Schrodinger equations with derivative nonlinearity Invited
平山 浩之
九州関数方程式セミナー
Event date： 2023.5.19
Presentation type：Oral presentation (general)
Awards 【 display / non-display 】
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Outstanding Contribution in Reviewing
2018.7 ELSEVIER Nonlinear Analysis
Hiroyuki Hirayama
Award type：Honored in official journal of a scientific society, scientific journal Country：Japan
Grant-in-Aid for Scientific Research 【 display / non-display 】
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パラメーターを含む非線形分散型方程式の連立系に対する時間大域的可解性について
Grant number：21K13825 2021.04 - 2025.03
独立行政法人日本学術振興会 科学研究費補助金 若手研究
Authorship：Principal investigator
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Lie 群構造をもつ非線形発展方程式の可解性の解明
Grant number：21K03333 2021.04 - 2024.03
独立行政法人日本学術振興会 科学研究費補助金 基盤研究(C)
Authorship：Coinvestigator(s)
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複雑な共鳴構造を持つ非線形分散型方程式の可解性について
Grant number：17K14220 2017.04 - 2023.03
科学研究費補助金 若手研究(B)
Authorship：Principal investigator
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臨界指数のソボレフ空間における非線型分散型方程式の適切性の解明
Grant number：14J00069 2014.04 - 2015.10
科学研究費補助金 特別研究員奨励費
Authorship：Principal investigator
Other research activities 【 display / non-display 】
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第7回 PDE Workshop in Miyazaki
2024.01
研究集会「第7回 PDE Workshop in Miyazaki」を開催した。
https://www.cc.miyazaki-u.ac.jp/pde/ -
数学と現象：Mathematics and Phenomena in Miyazaki 2023
2023.11
研究集会「数学と現象：Mathematics and Phenomena in Miyazaki 2023」の世話人を行った。
https://www.cc.miyazaki-u.ac.jp/math/mpm/mpm2023/ -
第4回 大同大学 若手微分方程式セミナー
2023.08
研究集会「第4回 大同大学 若手微分方程式セミナー」を開催した。
https://sites.google.com/view/daidowrks -
第6回 PDE Workshop in Miyazaki
2023.01
研究集会「第6回 PDE Workshop in Miyazaki」を開催した。
https://www.cc.miyazaki-u.ac.jp/pde/ -
第3回 大同大学 若手微分方程式セミナー
2022.08
研究集会「第3回 大同大学 若手微分方程式セミナー」を開催した。
https://www.daido-news.jp/du_news/notice/2986.html