非コンパクトな変分構造に付随する放物型・楕円型方程式の研究

• 研究課題/領域番号: 23540232
• 研究種目: 基盤研究(C)
• 配分区分: 基金
• 応募区分: 一般
• 研究分野: 大域解析学
• 研究機関: 福島大学
• 研究代表者: 石渡 通徳 福島大学, 共生システム理工学類, 准教授 (30350458)
• 研究期間 (年度): 2011 – 2013
• 研究課題ステータス: 完了 (2013年度)
• 配分額: 4,420千円 (直接経費: 3,400千円、間接経費: 1,020千円); 2013年度: 1,300千円 (直接経費: 1,000千円、間接経費: 300千円); 2012年度: 1,300千円 (直接経費: 1,000千円、間接経費: 300千円); 2011年度: 1,820千円 (直接経費: 1,400千円、間接経費: 420千円)
• キーワード: 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 / 変分問題 / コンパクト性の破れ / 臨界型関数不等式 / 国際情報交換 / 臨界問題 / 半線型放物型方程式 / 爆発解析

研究概要

本研究では非コンパクトな変分構造に付随する放物型方程式の時間大域解の漸近挙動、楕円型方程式の解の多重性、及び関連する臨界型関数不等式とそれに付随する最小化型変分問題の最小化元の存在・非存在を扱った。
放物型方程式については、臨界ソボレフ指数を非線型項の増大度にもつ方程式の時間大域解の挙動について、エネルギー―汎関数の値が量子化されることを、また楕円型方程式については、外部領域で定義された重み関数つきの方程式について、重み関数の空間無限遠での減衰が十分ゆるいときに多重解が存在することを明らかにした。関数不等式については、Trudinger-Moser 型不等式、Hardy 型不等式について論じた。

研究成果

• [雑誌論文] On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form (2014)
  • 著者名/発表者名: Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade
  • 雑誌名: Ann. I. H. Poincare 巻: 31, Issue 2 ページ: 297-314
  • 査読あり
• [雑誌論文] On the existence and nonexistence of maximizers associated with Trudinger-Moser type inequalities in unbounded domains (2013)
  • 著者名/発表者名: Michinori Ishiwata
  • 雑誌名: Emerging topics on differential equations and their applications, Nankai series in pure, applied mathematics and theoretical physics 巻: 10 ページ: 41-53
  • 査読あり
• [雑誌論文] Positive solutions to a semilinear parabolic equation associated with the critical Sobolev exponent (2013)
  • 著者名/発表者名: Michinori Ishiwata, Takashi Suzuki
  • 雑誌名: Nonlinear Differential Equations and Applications NoDEA 巻: 20, Issue 4 ページ: 1553-1576
  • 査読あり
• [雑誌論文] Multiple positive solutions of semilinear elliptic equations with weight function in exterior domain s (2013)
  • 著者名/発表者名: Michinori Ishiwata
  • 雑誌名: Differential and Integral Equations 巻: 26, Numbers 1-2 ページ: 183-200
  • 査読あり
• [雑誌論文] On the existence and nonexistence of maximizers associated with Trudinger-Moser type inequalities in unbounded domains (2013)
  • 著者名/発表者名: Michinori Ishiwata
  • 雑誌名: Emerging topics on differential equations and their applications 巻: 10 ページ: 41-53
  • 査読あり
• [雑誌論文] Multiple positive solutions of semilinear elliptic equations with weight function in exterior domains (2013)
  • 著者名/発表者名: Michinori Ishwiata
  • 雑誌名: Differential and Integral Equations 巻: 26 ページ: 183-200
  • 査読あり
• [雑誌論文] Heat equation with a singular potential on the boundary and the Kato inequality (2012)
  • 著者名/発表者名: Kazuhiro Ishige, Michinori Ishiwata
  • 雑誌名: Journal d'Analyse Mathematique 巻: 118, Issue 1 ページ: 161-176
  • 査読あり
• [雑誌論文] Global solutions for a semilinear heat equation in the exterior domain of a compact set (2012)
  • 著者名/発表者名: Kazuhiro Ishige, Michinori Ishiwata
  • 雑誌名: Discrete and continuous dynamical systems 巻: 32, Number 3 ページ: 847-866
  • 査読あり
• [雑誌論文] Global solutions for a semilinear heat equation in the exterior domain of a compact set (2012)
  • 著者名/発表者名: Kazuhiro Ishige and Michinori Ishiwata
  • 雑誌名: Discrete and continuous dynamical systems 巻: 32 ページ: 847-866
  • 査読あり
• [雑誌論文] Heat equation with a singular potential on the boundary and the Kato (2012)
  • 著者名/発表者名: Kazuhiro Ishige and Michinori Ishiwata
  • 雑誌名: Journal d'Analyse Math\'{e}matique 巻: 118 ページ: 161-176
  • 査読あり
• [雑誌論文] Existence and nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in R^N (2011)
  • 著者名/発表者名: Michinori Ishiwata
  • 雑誌名: Mathematische Annalen 巻: 351 ページ: 781-804
  • 査読あり
• [学会発表] The scale-invariant log-Hardy inequalities and related variational problems (2013)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 3rd Italian-Japanese workshop on geometric properties for parabolic and elliptic PDE's
  • 発表場所: Tokyo Institute of Technology, Tokyo, Japan
  • 年月日: 2013-09-06
• [学会発表] On the scale-invariant critical Hardy's inequality and related variational problems (2013)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: EQUADIFF 13
  • 発表場所: Faculty of Arts, Charles University, Prague, Czech Republic
  • 年月日: 2013-08-26
• [学会発表] On the global compactness type result for weighted energy functional involving Sobolev critical exponent and its applications (2013)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 南大阪応用数学セミナー
  • 発表場所: 大阪市立大学, 大阪
  • 年月日: 2013-04-27
• [学会発表] Semilinear parabolic equation with critical Sobolev exponent (2013)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: International Conference Quantization Blow-up and Concentration in Mathematical Physics View Point
  • 発表場所: Sigma Hall, Osaka University, Osaka, Japan
  • 年月日: 2013-03-27
• [学会発表] Variational problems associated with Trudinger-Moser inequalities in unbounded domains (2012)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: Swiss-Japanese Seminar
  • 発表場所: University of Zurich, Zurich, Swiss
  • 年月日: 2012-12-17
• [学会発表] Variational problems associated with Trudinger-Moser inequalities in unbounded domains (2012)
  • 著者名/発表者名: Michinori Ishiwata
  • 発表場所: University of Milan, Italy
  • 年月日: 2012-12-12
• [学会発表] Asymptotic behavior of time-global solutions for semilinear heat equation involving critical Sobolev exponent (2012)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 5th Polish-Japanese Days on Nonlinear Analysis in Interdisciplinary Sciences - Modellings, Theory and Simulations -
  • 発表場所: Kansai seminar house, Kyoto, Japan
  • 年月日: 2012-11-08
• [学会発表] Asymptotic behavior of Palais-Smale sequences in the critical problem and its application to semilinear heat equation (2012)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 5th Euro-Japanese Workshop on Blow-up, Centre International de Rencontres Mathematiques
  • 発表場所: Luminy, France
  • 年月日: 2012-09-10
• [学会発表] On the concentration and the leaking phenomena for solutions of reaction-diffusion equations (2012)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 応用数学-WPI-AIMR Math Unit 共催セミナー
  • 発表場所: 東北大学, 仙台
  • 年月日: 2012-07-12
• [学会発表] On the asymptotic behavior of solutions for semiliear parabolic equations involving critical Sobolev exponent (2012)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: The 9th AIMS conference on Dynamical systems, Differential equations and Applications
  • 発表場所: Hilton Grand Cypress, Orlando, FL, USA
  • 年月日: 2012-07-05
• [学会発表] Variational problems associated with Trudinger-Moser inequalities in unbounded domains (2012)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: The 9th AIMS conference on Dynamical systems, Differential equations and Applications
  • 発表場所: Hilton Grand Cypress, Orlando, FL, USA
  • 年月日: 2012-07-03
• [学会発表] Asymptotic behavior of solutions for semilinear parabolic equation with critical Sobolev exponent (2012)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: Regularity and Singularity for Geometric Partial Differential Equations and Conservation Laws
  • 発表場所: Resarch Institute of Mathematical Science, Kyoto university, Kyoto, Japan
  • 年月日: 2012-06-14
• [学会発表] Variational problems associated with Trudinger-Moser type inequalities in unbounded domains (2012)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: CONFERENCE ON Evolution Equations, Related Topics and Applications
  • 発表場所: Waseda university, Tokyo, Japan
  • 年月日: 2012-03-21
• [学会発表] スカラー場方程式の基底解の退化性について:基礎とその応用 (Part I, Part II) (2012)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 第2回室蘭非線形解析セミナー
  • 発表場所: 室蘭工業大学, 室蘭
• [学会発表] On the existence and nonexistence of maximizers associated with Trudinger-Moser type inequalities in unbounded domains (2011)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: Emerging Topics on Differential Equations and their Applications--Sino-Japan Conference of Young Mathematicians, Chern institute, Nankai University
  • 発表場所: Tianjin, People's Republic of China
  • 年月日: 2011-12-06
• [学会発表] Decay estimates for the Laplace equation with dynamical boundary condition and its application to haomonic extension inequality (2011)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 北九州地区における偏微分方程式研究集会
  • 発表場所: 九州工業大学, 小倉
  • 年月日: 2011-11-26
• [学会発表] The existence and the nonexistence of maximizing functions for the variational problems associated with Trudinger-Moser type inequalities in the whole domain (2011)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: MSJ-SI 2011
  • 発表場所: Kyusyu university, Hakata, Japan
  • 年月日: 2011-09-20
• [学会発表] Decay estimates for a solution to the Laplace equation with a dynamical boundary condition and related topics (2011)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: Second Italian-Japanese Workshop on geometric properties for parabolic and elliptic PDE's
  • 発表場所: Cortona, Italy
  • 年月日: 2011-06-22
• [学会発表] Decay estimates for a solution to the Laplace equation with a dynamical boundary condition and applications (2011)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 第525回「応用解析」研究会
  • 発表場所: 早稲田大学, 早稲田
  • 年月日: 2011-06-04
• [学会発表] On the existence and nonexistence of maximizers associated with Trudinger-Moser type inequalities in unbounded domains (2011)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: Sino-Japan Conference of Young Mathematicians(招待講演)
  • 発表場所: Nankai University
• [学会発表] The existence and the nonexistence of maximizing functions for the variational problems associated with Trudinger-Moser type inequalities in the whole domain (2011)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: MSJ-SI 2011(招待講演)
  • 発表場所: Kyusyu university
• [学会発表] Decay estimates for a solution to the Laplace equation with a dynamical boundary condition and related topics (2011)
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: GEOMETRIC PROPERTIES FOR PARABOLIC AND ELLIPTIC PDE's(招待講演)
  • 発表場所: Cortona
• [学会発表] The scale-invariant log-Hardy inequalities and related variational problems
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 3rd Italian-Japanese workshop on geometric properties for parabolic and elliptic PDE's
  • 発表場所: Tokyo Institute od Technology
  • 招待講演
• [学会発表] On the scale-invariant critical Hardy's inequality and related variational problems
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: EQUADIFF 13
  • 発表場所: Charles University, Czech Republic
  • 招待講演
• [学会発表] On the global compactness type result for weighted energy functional involving Sobolev critical exponent and applications
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 南大阪応用数学セミナー
  • 発表場所: 大阪市立大学
  • 招待講演
• [学会発表] Scale invariant version of the critical Hardy inequality and the associated variational problems
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 金沢解析セミナー
  • 発表場所: 金沢大学
  • 招待講演
• [学会発表] Semilinear parabolic equation with critical Sobolev exponent
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: International Conference Quantization Blow-up and Concentration
  • 発表場所: Sigma Hall, Osaka University, Osaka, Japan
  • 招待講演
• [学会発表] Variational problems associated with Trudinger-Moser
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: Swiss-Japanese Seminar
  • 発表場所: University of Z\"{u}rich, Z\"{u}rich, Swiss
  • 招待講演
• [学会発表] Asymptotic behavior of time-global solutions for semilinear heat equation involving critical Sobolev exponent
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 5th Polish-Japanese Days on Nonlinear Analysis in Interdisciplinary Sciences
  • 発表場所: Kansai seminar house, Kyoto, Japan
  • 招待講演
• [学会発表] Asymptotic behavior of Palais-Smale sequences in the critical problem and its application to semilinear heat equation
  • 著者名/発表者名: Michinori Ishiwata
  • 学会等名: 5th Euro-Japanese Workshop on Blow-up
  • 発表場所: Centre International de Rencontres Mathematiques, Luminy, France
  • 招待講演