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Norisuke Ioku

Section

Graduate School of Science

Job title

Associate Professor

Degree

PhD (Tohoku University)

researchmap

https://researchmap.jp/7000014872

J-GLOBAL ID

201601000052671268

e-Rad No.

50624607

Committee Memberships 1

日本数学会　函数方程式論分科会情報委員会運営員

2016/03 - 2020/03

Professional Memberships 1

THE MATHEMATICAL SOCIETY OF JAPAN

Research Areas 1

Natural sciences / Mathematical analysis /

Awards 2

福原賞

2022/12　日本数学会函数方程式論分科会　臨界型函数不等式に付随する変分問題および一般化された藤田型方程式の研究
日本数学会賞建部賢弘奨励賞

2014/09　日本数学会　対数型特異性にかかわる偏微分方程式の調和解析的研究

Papers 29

Structure of radial solutions to Hénon type equation on the hyperbolic space

Norisuke Ioku, Akira Toyoshima

JOURNAL OF DIFFERENTIAL EQUATIONS　453　2026/02/05

DOI： 10.1016/j.jde.2025.113828 　

ISSN： 0022-0396

eISSN： 1090-2732
P\'{o}lya--Szeg\H{o} type inequality for the Fourier rearrangement and comparison results for the fractional Laplacian Peer-reviewed

Norisuke Ioku, Tetsuya Yamamoto

Proceedings of the international conference ``Critical Phenomena in Nonlinear Partial Differential Equations, Harmonic Analysis, and Functional Inequalities''　2026
Singular solutions of semilinear elliptic equations with exponential nonlinearities in 2-dimensions Peer-reviewed

Yohei Fujishima, Norisuke Ioku, Bernhard Ruf, Elide Terraneo

Journal of Functional Analysis　289　(1)　110922-110922　2025/07
Publisher: Elsevier BV
DOI： 10.1016/j.jfa.2025.110922 　

ISSN： 0022-1236
Existence of solutions to a fractional semilinear heat equation in uniformly local weak Zygmund-type spaces Peer-reviewed

Norisuke Ioku, Kazuhiro Ishige, Tatsuki Kawakami

Analysis & PDE　18　(6)　1477-1510　2025/05/29
Publisher: Mathematical Sciences Publishers
DOI： 10.2140/apde.2025.18.1477 　

ISSN： 2157-5045

eISSN： 1948-206X
Brezis–Van Schaftingen–Yung formula in Orlicz spaces Peer-reviewed

Norisuke Ioku, Kyosuke Shibuya

Journal of Mathematical Analysis and Applications　538　(2)　128350-128350　2024/10
Publisher: Elsevier BV
DOI： 10.1016/j.jmaa.2024.128350 　

ISSN： 0022-247X
$W^{1,p}$ approximation of the Moser--Trudinger inequality Peer-reviewed

Masato Hashizume, Norisuke Ioku

Proceedings of the American Mathematical Society　2023
Quasi self-similarity and its application to the global in time solvability of a superlinear heat equation Peer-reviewed

Yohei Fujishima, Norisuke Ioku

Nonlinear Analysis　2023
Global in time solvability for a semilinear heat equation without the self-similar structure Peer-reviewed

Yohei Fujishima, Norisuke Ioku

Partial Differential Equations and Applications　3　(2)　2022/04

DOI： 10.1007/s42985-022-00158-3 　

ISSN： 2662-2963

eISSN： 2662-2971
Well-posedness of the cauchy problem for convection-diffusion equations in uniformly local Lebesgue spaces Peer-reviewed

Md Rabiul Haque, Norisuke Ioku, Takayoshi Ogawa, Ryuichi Sato

Differential and Integral Equations　34　(3-4)　223-244　2021

ISSN： 0893-4983
Solvability of a semilinear heat equation via a quasi scale invariance Peer-reviewed

Norisuke Ioku, Yohei Fujishima

GEOMETRIC PROPERTIES FOR PARABOLIC AND ELLIPTIC PDE's　2021
Non-uniqueness for a critical heat equation in two dimensions with singular data Peer-reviewed

Norisuke Ioku, Bernhard Ruf, Elide Terraneo

Annales de l'Institut Henri Poincar\'e/Analyse Non Lin\'eaire　36　2027-2051　2019
Attainability of the best Sobolev constant in a ball Peer-reviewed

Norisuke Ioku

Mathematische Annalen　375　(1-2)　1-16　2018/11
Critical dissipative estimate for a heat semigroup with a quadratic singular potential and critical exponent for nonlinear heat equations Peer-reviewed

Norisuke Ioku, Takayoshi Ogawa

Journal of Differential Equations　266　2274-2293　2018/08
Existence and nonexistence of solutions for the heat equation with a superlinear source term Peer-reviewed

Yohei Fujishima, Norisuke Ioku

Journal de Mathématiques Pures et Appliquées　(118)　128-158　2018
Hardy type inequalities in Lp with sharp remainders Peer-reviewed

Norisuke Ioku, Michinori Ishiwata, Tohru Ozawa

Journal of Inequalities and Applications　2017　(1)　5　2017/12/01
Publisher: Springer International Publishing
DOI： 10.1186/s13660-016-1271-1 　

ISSN： 1029-242X 1025-5834
Canceling effects in higher-order Hardy-Sobolev inequalities Peer-reviewed

Andrea Cianchi, Norisuke Ioku

Calculus of Variations and Partial Differential Equations　56　(2)　2017/02
L-p-L-q estimates for homogeneous operators Peer-reviewed

Norisuke Ioku, Giorgio Metafune, Motohiro Sobajima, Chiara Spina

COMMUNICATIONS IN CONTEMPORARY MATHEMATICS　18　(3)　2016/06

DOI： 10.1142/S0219199715500376 　

ISSN： 0219-1997

eISSN： 1793-6683
A Note on the Scale Invariant Structure of Critical Hardy Inequalities Peer-reviewed

Norisuke Ioku, Michinori Ishiwata

GEOMETRIC PROPERTIES FOR PARABOLIC AND ELLIPTIC PDE'S　176　(176)　97-120　2016

DOI： 10.1007/978-3-319-41538-3_7 　

ISSN： 2194-1009
Sharp remainder of a critical Hardy inequality Peer-reviewed

Norisuke Ioku, Michinori Ishiwata, Tohru Ozawa

ARCHIV DER MATHEMATIK　106　(1)　65-71　2016/01

DOI： 10.1007/s00013-015-0841-7 　

ISSN： 0003-889X

eISSN： 1420-8938
On a variational problem associated with a Hardy type inequality involving a mean oscillation Peer-reviewed

Norisuke Ioku, Michinori Ishiwata

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS　54　(4)　3949-3966　2015/12

DOI： 10.1007/s00526-015-0927-x 　

ISSN： 0944-2669

eISSN： 1432-0835
Sharp decay estimates in Lorentz spaces for nonnegative Schrodinger heat semigroups Peer-reviewed

Norisuke Ioku, Kazuhiro Ishige, Eiji Yanagida

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES　103　(4)　900-923　2015/04

DOI： 10.1016/j.matpur.2014.09.006 　

ISSN： 0021-7824

eISSN： 1776-3371
A Scale Invariant Form of a Critical Hardy Inequality Peer-reviewed

Norisuke Ioku, Michinori Ishiwata

INTERNATIONAL MATHEMATICS RESEARCH NOTICES　2015　(18)　8830-8846　2015

DOI： 10.1093/imrn/rnu212 　

ISSN： 1073-7928

eISSN： 1687-0247
Existence, Non-existence, and Uniqueness for a Heat Equation with Exponential Nonlinearity in R-2 Peer-reviewed

Norisuke Ioku, Bernhard Ruf, Elide Terraneo

MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY　18　(1)　2015

DOI： 10.1007/s11040-015-9199-0 　

ISSN： 1385-0172

eISSN： 1572-9656
Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation Peer-reviewed

Norisuke Ioku

JOURNAL OF FUNCTIONAL ANALYSIS　266　(5)　2944-2958　2014/03

DOI： 10.1016/j.jfa.2013.12.023 　

ISSN： 0022-1236

eISSN： 1096-0783
Sharp decay estimates of L-q-norms for nonnegative Schrodinger heat semigroups Peer-reviewed

Norisuke Ioku, Kazuhiro Ishige, Eiji Yanagida

JOURNAL OF FUNCTIONAL ANALYSIS　264　(12)　2764-2783　2013/06

DOI： 10.1016/j.jfa.2013.03.009 　

ISSN： 0022-1236
BREZIS-MERLE TYPE INEQUALITY FOR A HEAT EQUATION IN TWO DIMENSIONS Peer-reviewed

Norisuke Ioku

DIFFERENTIAL AND INTEGRAL EQUATIONS　24　(11-12)　1021-1036　2011/11

ISSN： 0893-4983
SOME SPACE-TIME INTEGRABILITY ESTIMATES OF THE SOLUTION FOR HEAT EQUATIONS IN TWO DIMENSIONS Peer-reviewed

Norisuke Ioku

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS　31　707-716　2011/09

ISSN： 1078-0947

eISSN： 1553-5231
The Cauchy problem for heat equations with exponential nonlinearity Peer-reviewed

Norisuke Ioku

JOURNAL OF DIFFERENTIAL EQUATIONS　251　(4-5)　1172-1194　2011/08

DOI： 10.1016/j.jde.2011.02.015 　

ISSN： 0022-0396
BREZIS-MERLE TYPE INEQUALITY FOR A WEAK SOLUTION TO THE N-LAPLACE EQUATION IN LORENTZ-ZYGMUND SPACES Peer-reviewed

Norisuke Ioku

DIFFERENTIAL AND INTEGRAL EQUATIONS　22　(5-6)　495-518　2009/05

ISSN： 0893-4983

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Misc. 4

書評「ジム・ヘンリー著，美味しい数学-証明の味はパイの味-」

猪奥倫左

数学通信　22　(3)　82-85　2017/11
Publisher: 日本数学会
ISSN： 1342-1387
Scale invariant structure of a critical Hardy inequality and its sharp remainder

IOKU Norisuke

2006　(2006)　12-19　2016/11
Publisher: 京都大学数理解析研究所
ISSN： 1880-2818
非斉次二次元熱方程式に対する振動型評価 (幾何学的偏微分方程式における保存則と正則性特異性の研究)

猪奥倫左

数理解析研究所講究録　1845　153-165　2013/07
Publisher: 京都大学
ISSN： 1880-2818
Hardy type inequalities with scale invariance in limiting cases (Progress in Variational Problems : Variational Problems Interacting with Probability Theories)

猪奥倫左

数理解析研究所講究録　1837　132-141　2013/06
Publisher: Kyoto University
ISSN： 1880-2818

Presentations 41

Attainability of the best Sobolev constant in a ball

猪奥倫左

弘前非線形方程式研究会　2019/11/23
Attainability of the best Sobolev constant in a ball

猪奥倫左

応用解析研究会　2019/11/16
凝スケール不変性を用いた半線型熱方程式の大域可解性について

猪奥倫左

数理物理と微分方程式　2019/11/01
Canceling effects in higher-order\\ Hardy-Sobolev inequalities

猪奥倫左

実解析学シンポジウム　2019/10/27
Attainability of the best Sobolev constant in a ball

猪奥倫左

実解析学シンポジウム　2019/10/27
凝スケール不変性を用いた半線型熱方程式の可解性について

猪奥倫左

京都大学NLPDEセミナー　2019/10/17
Solvability of a semilinear heat equation via a quasi scale invariance

猪奥倫左

応用数理解析セミナー　2019/10/16
Attainability of the best Sobolev constant in a ball

猪奥倫左

半田山微分方程式セミナー　2019/08/02
Attainability of the best Sobolev constant in a ball

猪奥倫左

愛媛大学解析セミナー　2019/06/16
Attainability of the best Sobolev constant in a ball

猪奥倫左

VI Italian-Japanese Workshop, Geometric properties for parabolic and elliptic PDEs　2019/05/22
Attainability of the best Sobolev constant in a ball

猪奥倫左

さいたま数理解析セミナー　2019/03/23
Attainability of the best Soev constantly in a ball

猪奥倫左

AMS Sectional Meeting　2019/03/23
Attainability of the best Sobolev constant in a ball

猪奥倫左

岐阜数理科学セミナー　2019/02/15
Sobolev不等式のスケール不変性と最良定数の達成可能性について

猪奥倫左

第7回室蘭連続講演会　2019/01/23
Attainability of the best Sobolev constant in a ball Invited

猪奥倫左

九州関数方程式セミナー　2018/12/14
Attainability of the best Sobolev constant in a ball Invited

猪奥倫左

広島大学HMAセミナー　2018/12/07
Attainability of the best Sobolev constant in a ball

IOKU Norisuke

2018/11
Attainability of the best Sobolev constants in a ball

猪奥倫左

日本数学会秋期総合分科会　2018/09
Critical dissipative estimate for a heat semigroup with the inverse square potential

猪奥倫左

日本数学会秋期総合分科会　2018/09
Remark on a Sobolev type inequality in a ball

IOKU Norisuke

AIMS 2018, Recent advances in the calculus of variations and elliptic PDE　2018/07
Canceling effects in higher-order Hardy-Sobolev inequalities

IOKU Norisuke

AMS Sectional Meeting, Special Session on Partial Differential Equations and New Perspective of Variational Methods　2018/04
On Talenti type function

IOKU Norisuke

Analysis seminar at Firenze University　2018/03
Remark on a Sobolev type inequality in the unit ball

IOKU Norisuke

Analysis seminar, Milano University　2018/03
Canceling effects in higher-order Hardy-Sobolev inequalities

IOKU Norisuke

International Conference Nonlinear Partial Differential Equations 2018　2018/02
高階Hardy-Sobolevの不等式が持つ相殺効果

猪奥倫左

第11回実解析と関数解析による偏微分方程式論　2017/12
半線形熱方程式の可解性の分類

愛媛大学における微分方程式セミナー　2017/08/31
Existence and nonexistence of solutions for the heat equation with a superlinear source term

Norisuke Ioku

Pacific Rim Mathematical Association PRIMA 2017　2017/08/15
半線形熱方程式の可解性の分類

九州関数方程式セミナー　2017/07/14
Existence and nonexistence of solutions for the heat equation with a superlinear source term

南大阪解析セミナー　2017/06
半線形熱方程式の可解性の分類

金沢解析セミナー　2017/06
Existence and nonexistence of solutions for the heat equation with a superlinear source term

International Conference on PDEs, Geometric Analysis and Functional Inequalities　2017/03
半線形熱方程式の可解性の分類

第10回実解析と関数解析による微分方程式セミナー　2016/12
Existence and Nonexistence of Solutions for the Heat Equation with a Superlinear Source Term

Korea-Japan International Workshop of Nonlinear Partial Differential Equations --Aspect of Regularity and Asymptotics--　2016/11
高階Hardy-Sobolevの不等式が持つ対数型特異性の相殺効果

第27回数理物理と微分方程式　2016/11
Existence and Nonexistence of Solutions for the Heat Equation with a Superlinear Source Term

Geometry of solutions of PDE's and its related reverse problem　2016/10
Canceling effects in higher-order Hardy-Sobolev inequalities

Partial Differential Equations and Related Topics　2016/09
Existence and nonexistence of solutions for a heat equation with a superlinear source term

11th AIMS conference　2016/07/04
The semilinear heat equation with an exponential nonlinearity

Geometric and Physical aspects of Trudinger-Moser type inequalities　2016/06/26
Existence and nonexistence of solutions for a heat equation with a superlinear source term

Geometric aspects of PDE’s and functional inequalities　2016/04/30
半線形熱方程式の可解性の分類

愛媛大学解析セミナー　2016/04
Existence and nonexistence of solutions for a heat equation with a superlinear source term

Analysis seminar in Firenze University　2016/02

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Research Projects 21

臨界型非線型偏微分方程式の非線型境界条件における臨界性の発見

小川卓克, 高橋太, 瀬片純市, 服部裕司, 猪奥倫左, 勝呂剛志, 中里亮介

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 基盤研究(A)

Institution: 早稲田大学

2025/04/01 - 2030/03/31
New development on higher order elliptic and parabolic PDEs -- cooperation between harmonic analysis and geometric analysis

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Fund for the Promotion of Joint International Research (Fostering Joint International Research (B))

Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research (B))

Institution: Tohoku University

2020/10/27 - 2025/03/31
q-指数関数と一般化三角関数を繋ぐオイラー型関係式

猪奥倫左

Offer Organization: 日本学術振興会

System: 科学研究費助成事業挑戦的研究(萌芽)

Category: 挑戦的研究(萌芽)

Institution: 東北大学

2021/07/09 - 2024/03/31

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オイラー型関係式を得るための状況証拠を探るべく，臨界型関数不等式を劣臨界の連続極限として導出する研究を行った．具体的には，Trudinger--Moser不等式を連続極限として持つ汎関数としてq-指数関数を選び，それに伴う劣臨界不等式が関数不等式としてTrudinger--Moser不等式に収束すること，および劣臨界集中レベルが臨界集中レベルに収束することを証明した．得られた結果は専門欧文雑誌に投稿中である．
Creation of advanced method in mathematical analysis on nonlinear mathematical models of critical type

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)

Category: Grant-in-Aid for Scientific Research (S)

Institution: Tohoku University

2019/06/26 - 2024/03/31
臨界型変分問題における領域の幾何の影響-解空間大域構造とコンパクト性喪失機構-

高橋太, 加藤信, 橋詰雅斗, 石渡通徳, 猪奥倫左, 佐野めぐみ, 高津飛鳥

Offer Organization: 日本学術振興会

System: 科学研究費助成事業基盤研究(B)

Category: 基盤研究(B)

2019/04/01 - 2023/03/31

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本研究課題では、Sobolev 不等式、Hardy 不等式などの関数不等式の最良定数を定める最小化問題や Trudinger-Moser 不等式に由来する変分問題など、その近似解の列の相対コンパクト性がアプリオリには期待できない「臨界型変分問題」を取り扱い、解空間の大域的構造、及び近似解の列がコンパクト性を喪失する機序について研究することを目的としている。特に本研究課題では、変分問題の解空間（エネルギー汎関数の臨界点の集合）の大域的構造や近似解の列の非コンパクト性が、領域の境界の曲率や形状、滑らかさなどの微分幾何学的性質にどのように影響されるのかを定量的に解明することを目指す。より具体的には、以下の課題について新しく結果を得ることを目的とする。(1) 種々の Trudinger-Moser 型不等式に付随する変分汎関数の臨界点集合の大域的構造と領域の微分幾何学的性質との相関、特に領域が凸な場合の最大化関数の一意性の成否(2) 種々の Hardy 型不等式に付随する最小化問題の最小化列のコンパクト性喪失メカニズムと領域の幾何との相関(3) 臨界変分構造を持つ種々の楕円型方程式に対する特異領域上での爆発解析本年度は制約条件付きベクトル場に対する Hardy-Leray 不等式の解析が大きく進展し、この題材での研究論文を3本公刊することができた。また、強い異方性を持つ Finsler ラプラス作用素を主部に持つ Hardy 不等式の基本的な結果を証明し、今後この結果は Finsler ラプラス作用素を主部に持つ楕円型方程式の解の安定性や爆発解析に利用されることが期待される。
Development of analysis method for critical problems with logarithmic singularity

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))

Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))

Institution: Tohoku University

2020 - 2022
Singularity of solutions and stationary problems for nonlinear parabolic equations

Naito Yuki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

Category: Grant-in-Aid for Scientific Research (C)

2017/04/01 - 2021/03/31

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We showed the existencde and the uniquness of the singular solutions to semilinear elliptic partial differential equations with Sobolev super-critical nonlinearity. We also showed the convergence of regular solutions to the singular solution. We consider positive solutions of the semilinear heat equation with supercritical power nonlinearity, and construct peaking solutions by connecting a backward selfsimilar solution with a forward self-similar solution. In particular, we show the existence of incomplete blow-up solutions with blow-up profile above the singular steady state.
対数型スケール変換が拓く関数不等式の新展開と偏微分方程式への応用 Competitive

猪奥倫左

Offer Organization: 日本学術振興会

System: 若手B

2018/04 - 2021/03
Unravel higher order critical structures to solutions of nonlinear dispersive and dissipative partial differential equations

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

Category: Grant-in-Aid for Scientific Research (A)

Institution: Tohoku University

2019/04/01 - 2020/03/31
Behavior of solutions for various nonlinear diffusion equations

Fujishima Yohei, IOKU Norisuke

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

Category: Grant-in-Aid for Young Scientists (B)

Institution: Shizuoka University

2015/04/01 - 2019/03/31

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We consider a nonlinear heat equation, which is an example of reaction-diffusion equations, and studied the solvability of the initial-value problem. In particular, we treated a nonlinear term which we do not assume a specific form, and derived the optimal integrability of initial function for the local in time existence of solutions. Furthermore, we also considered the heat equation with exponential nonlinearity, and obtained the optimal decay rate of initial function for global in time existence of solutions.
Critical exponent and the behavior of solutions to nonlinear parabolic partial differential equations

Yuki Naito, Yanagida Eiji, Ishwata Michinori, Senba Takasi, Kajikiya Ryuji, Yoshikawa Syuji, Ioku Norisuke

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

Category: Grant-in-Aid for Scientific Research (C)

Institution: Ehime University

2014/04/01 - 2018/03/31

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We consider the semilinear elliptic equation and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures. We show that, under the suitable conditions, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across the critical exponent which is determined by the space dimension and the order of the behavior of the coefficient function. We consider the behavior of solutions to the Cauchy problem for a semilinear heat equation with supercritical nonlinearity. We study the convergence of solutions to steady states in a weighted norm, and show the global attractivity property of steady states. We also give its convergence rate for a class of initial data. Proofs are given by a comparison method based on matched asymptotic expansion.
臨界Hardyの不等式と対数型特異性を伴う偏微分方程式への応用 Competitive

猪奥倫左

Offer Organization: 日本学術振興会若手B

2015/04 - 2018/03
関数不等式の最良定数とその周辺 Competitive

猪奥倫左

Offer Organization: 京都大学数理解析研究所

System: 2017年度RIMS共同研究（グループ型）

2017/09 - 2017/09
Explore of undiscovered variational principles in function spaces in real analysis

Takahashi Futoshi, IOKU Norisuke

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research

Category: Grant-in-Aid for Challenging Exploratory Research

Institution: Osaka City University

2014/04/01 - 2017/03/31

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The aim of this research is to understand the variational structures of several functional inequalities, such as Trudinger-Moser, Sobolev, and Hardy type, which are established newly in various functional spaces such as Lorentz, and Orlicz, and to find new applications to PDE theories.More precise research subjects are the following:(1) Sobolev-Orlicz approach to elliptic systems with the indefinite variational structures, (2) Study of the Trudinger-Moser type inequalities and their variational structures (3) Hardy type inequalities of the scale invariant form and its application to the stability theory of solutions.
臨界関数不等式に関わる諸問題が持つ不変構造の探求 Competitive

猪奥倫左

Offer Organization: 京都大学数理解析研究所

System: 2016年度RIMS共同研究

2017/02 - 2017/02
臨界Hardyの不等式の伸縮不変性とスペクトル理論への応用 Competitive

猪奥倫左

Offer Organization: 公益財団法人住友財団

2014/11 - 2016/10
Singularity of solutions for nonlinear partial differential equations of parabolic type and structure of solutions for the stationary problems

NAITO Yuki, KAJIKIYA Ryuji, ISHII Katsuyuki, YANAGIDA Eiji, SENBA Takasi, YOSHIKAWA Syuji, IOKU Norisuke

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

Category: Grant-in-Aid for Scientific Research (C)

Institution: Ehime University

2011/04/28 - 2015/03/31

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We study the singular behavior of solutions for nonlinear partial differential equations of parabolic type, and investigate the relations between the singularity and the solution structure of the stationary problems. We verify the roles of self-similar solutions in the Cauchy problems for semilinear heat equations in the case where the problem has multiple self-similar solutions. We consider the Cauchy problem for semilinear heat equations, and show the optimal spatial decay condition of initial functions at infinity for the blow-up in finite time. We consider the elliptic partial differential equations involving p-Laplace operator, and show the geometrical properties of radially symmetric solutions which has singular behavior near the boundary.
臨界Hardyの不等式の構造解明と偏微分方程式への応用 Competitive

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Offer Organization: 愛媛大学理学部長裁量研究助成

2014/04 - 2015/03
臨界函数不等式と偏微分方程式の正則性理論 Competitive

猪奥倫左

Offer Organization: 愛媛大学大学院理工学研究科共同研究支援経費

2012/04 - 2013/03
楕円型偏微分方程式に対する臨界正則性理論 Competitive

猪奥倫左

Offer Organization: 愛媛大学研究活性化事業スタートアップ支援

2011/10 - 2012/03
非線形楕円型, 放物型方程式の正則性と臨界不等式

猪奥倫左

Offer Organization: 日本学術振興会

System: 科学研究費助成事業特別研究員奨励費

Category: 特別研究員奨励費

2010 - 2011

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本年度も引き続き,楕円型,放物型偏微分方程式の正則性理論について研究した.正則性研究においては外力項を持つPoisson方程式の境界値問題が最も代表的である.先行研究において,解が満たす正則性は考察するEuclid空間の次元と外力項の可積分性条件に大きく依存することが知られている.特に空間が二次元で外力が可積分関数の場合には,基本解の特殊性から楕円型正則性が破綻するため超関数解は強解とならず,またSobolevの埋め込みの破綻が同時に起きるため解は弱解(エネルギークラスに属する解)にもならない.この意味で空間二次元,外力可積分条件の場合における正則性研究は二重臨界問題と呼ばれており,次の二つの方向に研究が進んでいる.一つはBrezis-Merleによる指数可積分性を用いた正則性の表現であり,もう一つは解の平均振動の有界性(BMO評価)を用いた正則性の表現である. 以上の背景を踏まえ,本年度の研究においては,前者の可積分性を用いた方法に着目し,臨界問題の高次元化であるN-Laplace方程式,臨界問題の時間発展化である二次元熱方程式の正則性問題を研究した.高次元化問題においては,対応する非臨界問題との相違点を明確にした.また,時間発展化問題においては空間振動有界性,時間についての有界性評価を最良定数込みで求めた.熱方程式の解は外力項が時間に依存しない場合,定常問題の解に収束することが知られているため,この結果は定常問題の拡張になっている. また,これまでの臨界正則性を更に発展させるため,ポテンシャル項付きの臨界問題(Hardyの不等式)の研究を開始した.本研究では特に,既存の有界領域における結果を自然に全空間に拡張することを考え,対数補正項を用いない形での臨界正則性を部分的に得た.

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愛媛大学理学部公開講座

2017/11/11 -