A study on the asymptotic behavior of infinite-time blow up solutions for parabolic equations involving critical Hardy-Sobolev exponent

• Project/Area Number: 19740081
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Muroran Institute of Technology
• Principal Investigator: ISHIWATA Michinori Muroran Institute of Technology, 共生システム理工学類, 准教授 (30350458)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥3,860,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥660,000); Fiscal Year 2010: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥1,000,000 (Direct Cost: ¥1,000,000)
• Keywords: 臨界指数 / 漸近挙動 / コンパクト性の破れ / 無限時間爆発 / 臨界性 / トレース・ハーディ不等式 / 力学的境界条件 / エネルギーの量子化 / 関数方程式の大域理論 / ハーディの不等式 / トレース作用素 / ソボレフの不等式 / 藤田型方程式 / 無限時問爆発 / 放物型方程式 / 非線型偏微分方程式 / 非線形微分方程式

Research Abstract

In this study, we are concerned with the profiles of infinite time blow-up solutions for semilinear parabolic equations involving critical Sobolev-Hardy exponents. For the critical Sobolev case, which is a special case of the critical Sobolev-Hardy case, we find that the asymptotic behavior of infinite time blow-up solutions are governed by the nontrivial stationary solutions of the equations. We also treat the several parabolic equations which are related with Hardy and Sobolev inequalities.

Research Products

• [Journal Article] On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent (2010)
  • Author(s): Michinori Ishiwata
  • Journal Title: J.Differential Equations 249 No.6 Pages: 1466-1482
  • Peer Reviewed
• [Journal Article] Semilinear parabolic equation in \R^N associated with critical Sobolev exponent (2010)
  • Author(s): Ryo Ikehata, Michinori Ishiwata, Takashi Suzuki
  • Journal Title: Annales de l'Institut Henri Poincare - Analyse non lineaire 27 no.3 Pages: 877-900
  • Peer Reviewed
• [Journal Article] On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent (2010)
  • Author(s): Michinori Ishiwata
  • Journal Title: J.Differential Equations Volume: 249 Pages: 1466-1482
  • Peer Reviewed
• [Journal Article] Semilinear parabolic equation in $￥R^N$ associated with critical Sobolev exponent (2010)
  • Author(s): Ryo Ikehata, Michinori Ishiwata, Takashi Suzuki
  • Journal Title: Annales de l'Institut Henri Poincare-Analyse nonblineaire Volume: 27 Pages: 877-900
  • Peer Reviewed
• [Journal Article] The decay of the solutions for the heat equation with a potential (2009)
  • Author(s): Kazuhiro Ishige, Michinori Ishiwata, Tatsuki Kawakami
  • Journal Title: Indiana University Mathematics Journal 58 No.6 Pages: 2673-2708
  • Peer Reviewed
• [Journal Article] Effect of topology on the multiplicity of solutions for some semilinear elliptic systems with critical Sobolev exponent (2009)
  • Author(s): Michinori Ishiwata
  • Journal Title: Nonlinear Differ.Equ.Appl. 16 Pages: 283-296
  • Peer Reviewed
• [Journal Article] Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in \R^2 (2009)
  • Author(s): Michinori Ishiwata, Takayoshi Ogawa, Futoshi Takahashi
  • Journal Title: Electron.J.Diff.Eqns. Vol.2009 No.32 Pages: 1-10
  • Peer Reviewed
• [Journal Article] The decay of the solutions for the heat equation with a potential (2009)
  • Author(s): Kazuhiro Ishige, Michinori Ishiwata, Tatsuki Kawakami
  • Journal Title: Indiana University Mathematics Journal 58,No.6 Pages: 2673-2708
• [Journal Article] Effect of topology on the multiplicity of solutions for some semilinear elliptic systems with critical Sobolev exponent (2009)
  • Author(s): Ishiwata Michinori
  • Journal Title: Nonlinear Differ. Equ. Appl. 16 Pages: 283-296
• [Journal Article] Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in R2 (2009)
  • Author(s): Michinori Ishiwata, Takayoshi Ogawa, Futoshi Takahashi
  • Journal Title: Electron. J. Diff. Eqns. Vol.2009,No.32 Pages: 1-10
• [Journal Article] The decay of solutions for the heat equation with a potential (2009)
  • Author(s): Kazuhiro Ishige, Michinori Ishiwata, Tastuki Kawakami
  • Journal Title: Indiana University Mathematics Journal 58 Pages: 2673-2708
  • Peer Reviewed
• [Journal Article] Effect of topology on the multiplicity of solutions for some semilinear elliptic systems with critical Sobolev exponent (2009)
  • Author(s): Michinori Ishiwata
  • Journal Title: Nonlinear Differential Eq.and Applications 16 Pages: 283-296
  • Peer Reviewed
• [Journal Article] Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in R2 (2009)
  • Author(s): Michinori Ishiwata, Takayoshi Ogawa, Futoshi Takahashi
  • Journal Title: Electron.J.Diff.Eqns 2009, No.32 Pages: 1-10
  • Peer Reviewed
• [Journal Article] Existence of multiple solutions for nonlinearly perturbed elliptic parabolic system in R^2 (2009)
  • Author(s): Michinori Ishiwata, Futoshi Takahashi, Takayoshi Ogawa
  • Journal Title: Electronic journal of differential equations 32 Pages: 1-10
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of strong solutions for nonlinear parabolic equations with critical Sobolev exponent (2008)
  • Author(s): Michinori Ishiwata
  • Journal Title: Advances in Differential Equations 13(3-4) Pages: 349-366
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of strong solutions for nonlinear parabolic equations with critical Sobolev exponent (2008)
  • Author(s): Michinori Ishiwata
  • Journal Title: Advances in Differential Equations 13 Pages: 349-366
  • Peer Reviewed
• [Journal Article] Multiple solutions for semilinear elliptic systems involving critical Sobolev exponent (2007)
  • Author(s): Michinori Ishiwata
  • Journal Title: Differential and Integral Equations 20(11) Pages: 1237-1252
  • Peer Reviewed
• [Journal Article] On bounds for global solutions of semilinear parabolic equations with critical and subcritical Sobolev exponent (2007)
  • Author(s): Michinori Ishiwata
  • Journal Title: Differential and Integral Equations 20(9) Pages: 1021-1034
  • Peer Reviewed
• [Journal Article] Multiple solutions for semilinear elliptic systems involving critical Sobolev exponent (2007)
  • Author(s): Michinori Ishiwata
  • Journal Title: Differential and Integral Equations 20 Pages: 1237-1252
  • Peer Reviewed
• [Journal Article] On bounds for global solutions of semilinear parabolic equations with critical and subcritical Sobolev exponent (2007)
  • Author(s): Michinori Ishiwata
  • Journal Title: Differential and Integral Equations 20 Pages: 1021-1034
  • Peer Reviewed
• [Journal Article] Existence and nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in \R^N
  • Author(s): Michinori Ishiwata
  • Journal Title: Math. Ann., to appear.
  • Peer Reviewed
• [Presentation] Asymptotic behavior of solutions for semilinear parabolic equations involving critical Sobolev exponent : a variational point of view (2010)
  • Author(s): Michinori Ishiwata
  • Organizer: 非線形問題に現れる特異性の解析(SNP2010)
  • Place of Presentation: Kansai seminar house, Kyoto, Japan.
  • Year and Date: 2010-11-29
• [Presentation] Asymptotic behavior of solutions for semilinear parabolic equations involving critical Sobolev exponent : a variational point of view (2010)
  • Author(s): Michinori Ishiwata
  • Organizer: 非線形問題に現れる特異性の解析(SNP2010)
  • Place of Presentation: 関西セミナーハウス
  • Year and Date: 2010-11-29
• [Presentation] Trace type Hardy inequalities in the half space and related topics (2010)
  • Author(s): Michinori Ishiwata
  • Place of Presentation: University of Naples, Italy.
  • Year and Date: 2010-11-02
• [Presentation] On the asymptotic behavior of solutions for semilinear parabolic equations involving critical Sobolev exponent II (2010)
  • Author(s): 石渡通徳
  • Organizer: 広島微分方程式研究会
  • Place of Presentation: 広島大学,西条
  • Year and Date: 2010-10-08
• [Presentation] Asymptotic behavior of solutions for semilinear parabolic equations involving critical Sobolev exponent (2010)
  • Author(s): Michinori Ishiwata
  • Organizer: 4th Euro-Japanese Workshop on Blow-up
  • Place of Presentation: Lorentz center, Leiden, Netherland.
  • Year and Date: 2010-09-10
• [Presentation] Asymptotic behavior of solutions for semilinear parabolic equations involving critical Sobolev exponent (2010)
  • Author(s): Michinori Ishiwata
  • Organizer: 4th Euro-Japanese Workshop on Blow-up
  • Place of Presentation: Lorentz center, Netherland
  • Year and Date: 2010-09-10
• [Presentation] On the variational problems associated with Trudinger-Moser type inequalties in unbounded domains (2010)
  • Author(s): Michinori Ishiwata
  • Organizer: The 8th AIMS donference on Dynamical systems, Differential equations and Applications
  • Place of Presentation: Dresden university of technology, Dresden, German.
  • Year and Date: 2010-05-25
• [Presentation] On the variational problems associated with Trudinger-Moser type inequalties in unbounded domains (2010)
  • Author(s): Michinori Ishiwata
  • Organizer: The 8th AIMS conference on Dynamical systems, Differential equations and Applications
  • Place of Presentation: Dresden university of technology
  • Year and Date: 2010-05-25
• [Presentation] L^p-L^q type estimates for Laplace equations with dynamical boundary conditions (2010)
  • Author(s): 石渡通徳
  • Organizer: 第18回「応用解析」研究会
  • Place of Presentation: 南風荘,箱根
  • Year and Date: 2010-03-01
• [Presentation] On the dynamical systems with noncompact orbit, RIMS 共同研,Research on the elliptic and the parabolic equations involving the energy functional with lack of compactness (2010)
  • Author(s): 石渡通徳
  • Organizer: 京都大学数理解析研究所
  • Place of Presentation: 京都
  • Year and Date: 2010-02-17
• [Presentation] On the maximizing problem associated with Trudinger-Moser type inequalities in \R^2 (2010)
  • Author(s): 石渡通徳
  • Organizer: 松山解析セミナー2010
  • Place of Presentation: 愛媛大学,松山
  • Year and Date: 2010-02-05
• [Presentation] On the existence and nonexistence of maximizers for variational problems asociated with Trudinger-Moser type inequalities in unbouded domains (2010)
  • Author(s): 石渡通徳
  • Organizer: 微分方程式深江ワークショップ
  • Place of Presentation: 神戸大学,神戸
  • Year and Date: 2010-01-22
• [Presentation] Asymptotic behavior of solutions for semilinear parabolic equations involving critical Sobolev exponent (2009)
  • Author(s): Michinori Ishiwata
  • Place of Presentation: University of Florence, Italy.
  • Year and Date: 2009-11-06
• [Presentation] Asymptotic behavior of solutions for semilinear parabolic equations involving critical Sobolev exponent (2009)
  • Author(s): Michinori Ishiwata
  • Place of Presentation: University of Naples, Italy.
  • Year and Date: 2009-11-03
• [Presentation] Existence of maximizing functions for functionals of critical growth (2009)
  • Author(s): Michinori Ishiwata
  • Organizer: 1^<st> Italian-Japanese workshop on geometric properties for parabolic and elliptic PDE's
  • Place of Presentation: Tohoku university
  • Year and Date: 2009-06-16
• [Presentation] Exsitence of maximizers for functionals of critical growth (2009)
  • Author(s): 石渡通徳
  • Organizer: 変分問題とその周辺
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-06-10
• [Presentation] The asymptotic behavior of solutions of semilinear parabolic equation with critical Sobolev exponent (2009)
  • Author(s): Michinori Ishiwata
  • Organizer: 4th Polish - Japanese Days Current advances in applied nonlinear analysis and mathematical modelling issues
  • Place of Presentation: Warsaw/Madralin, Poland.
  • Year and Date: 2009-05-20
• [Presentation] The asymptotic behavior of solutions of semilinear parabolic equation with critical Sobolev exponent (2009)
  • Author(s): Michinori Ishiwata
  • Organizer: $th Polish-Japanese days current advances in applied nonlinear analysis
  • Place of Presentation: ICM, Warsaw, Poland
  • Year and Date: 2009-05-19
• [Presentation] 臨界ソボレフ指数をもつ半線型熱方程式:変分法的立場から (2009)
  • Author(s): 石渡通徳
  • Organizer: 2009年度日本数学会・春季総合分科会・函数方程式分科会特別講演
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-26
• [Presentation] 臨界ソボレフ指数をもつ半線型熱方程式 : 変分法的立場から (2009)
  • Author(s): Michinori Ishiwata
  • Organizer: 日本数学会
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-26
• [Presentation] On the variational problems arising from Trudinger-Moser type ineaulaities in unbounded domains (2009)
  • Author(s): Michinori Ishiwata
  • Organizer: Workshop on Variational Problems and Elliptic Equations
  • Place of Presentation: Osaka university
  • Year and Date: 2009-03-09
• [Presentation] Existence of an extremal function for Trudinger-Moser type inequalities in the whole domain (2009)
  • Author(s): 石渡通徳
  • Organizer: 第16回応用解析研究会シンポジウム
  • Place of Presentation: 熱海
  • Year and Date: 2009-03-03
• [Presentation] Variational problems associated with Trudinger-Moser type inequalities in \R^N (2009)
  • Author(s): 石渡通徳
  • Organizer: 確率論とPDE
  • Place of Presentation: 広島大学,広島
  • Year and Date: 2009-01-20
• [Presentation] Exsitence of maximizing functions for functionals of critical growth (2009)
  • Author(s): Michinori Ishiwata
  • Organizer: 1st Italian-Japanese workshop on geometric properties for parabolic and elliptic PDE's
  • Place of Presentation: Tohoku University, Sendai, Japan.
• [Presentation] Exsitence of maximizing functions for functionals of critical growth (2009)
  • Author(s): Michinori Ishiwata
  • Organizer: 1st Italian-Japanese workshop on geometric properties for parabolic and elliptic PDE's
  • Place of Presentation: Tohoku University, Sendai, Japan
• [Presentation] Potential well type results for critical semilinear parabolic equations with gradient term (2008)
  • Author(s): 石渡通徳
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-26
• [Presentation] Potentiall well type results for critical semilinear parabolic equations with gradient term (2008)
  • Author(s): Michinori Ishiwata
  • Organizer: 日本数学会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-26
• [Presentation] Existece of a maximizer for Trudinger-Moser type inequalities in \Bbb R^2 (2008)
  • Author(s): 石渡通徳
  • Organizer: 第四回 非線型の諸問題
  • Place of Presentation: 佐賀大学,佐賀
  • Year and Date: 2008-09-22
• [Presentation] On the threshold solutions for semilinear parabolic problems involving critical Sobolev exponent (2008)
  • Author(s): Ishiwata Michinori
  • Organizer: Third Euro-Japanese workshop on blow-up
  • Place of Presentation: Tohoku University, Sendai, Japan.
  • Year and Date: 2008-09-08
• [Presentation] Asymptotic behavior of time global solutions for semilinear parabolic equation with critical Sobolev exponent (2008)
  • Author(s): 石渡通徳
  • Organizer: 第2回岐阜非線形ワークショップ-非線形現象の数理解析-
  • Place of Presentation: 岐阜大学,岐阜
  • Year and Date: 2008-07-31
• [Presentation] On the concentration phenomena of solutions for semilinear parabolic equations involving critical Sobolev exponent (2008)
  • Author(s): Ishiwata Michinori
  • Organizer: Ryukoku Workshop 2008, Recent Progress on Pattern Formation and Dynamics in Mathematical Sciences
  • Place of Presentation: Ryukoku university, Kyoto, Japan.
  • Year and Date: 2008-06-13
• [Presentation] A remark on the asymptotic behavior of solutions for critical parabolic problems II (2008)
  • Author(s): 石渡通徳
  • Organizer: 日本数学会春季総合分科会
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-25
• [Presentation] 変分解析入門 (2008)
  • Author(s): 石渡通徳
  • Organizer: サーベイレクチャーシリーズ「非線形問題の数学解析」
  • Place of Presentation: 東北大学,仙台
• [Presentation] Formation of singularities in time-global solutions for critical semilinear parabolic equations in \Bbb{R}^N (2007)
  • Author(s): 石渡通徳
  • Organizer: 第5回浜松偏微分方程式研究集会
  • Place of Presentation: 静岡大学
  • Year and Date: 2007-12-17
• [Presentation] Bubbling phenomena for semilinear parabolic equations involving critical Sobolev exponent (2007)
  • Author(s): 石渡通徳
  • Organizer: 非線形解析学の手法による数理物理学の研究II
  • Place of Presentation: 東北大学
  • Year and Date: 2007-12-12
• [Presentation] Asymptotic behavior of time-global solutions for a semilinear parabolic equation involving critical Sobolev exponent (2007)
  • Author(s): Ishiwata Michinori
  • Organizer: First Chile-Japan workshop on Nonlinear Elliptic and Parabolic PDE
  • Place of Presentation: Centro de modelamiento mathematico, departmento de ingenieria matematica, universidad de Chile, Santiago, Chile.
  • Year and Date: 2007-10-23
• [Presentation] On the asymptotic behavior of solutions for semilinear parabolic equations with critical Sobolev exponent (2007)
  • Author(s): 石渡通徳
  • Organizer: 研究集会「広島微分方程式研究会」
  • Place of Presentation: 広島
  • Year and Date: 2007-10-13
• [Presentation] A remark on the asymptotic behavior of solutions for critical parabolic problems (2007)
  • Author(s): 石渡通徳
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-24
• [Presentation] A remark on the asymptotic behavior of solutions for critical parabolic problem (2007)
  • Author(s): Michinori Ishiwata
  • Organizer: 2007年度日本数学会秋季総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-24
• [Presentation] Multiple existence of positive solutions for semilinear elliptic systems with critical Sobolev exponent (2007)
  • Author(s): 石渡通徳
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-21
• [Presentation] Multiple existence of positive solutions for semilinear elliptic systems with critical Sobolev exponent (2007)
  • Author(s): Michinori Ishiwata
  • Organizer: 2007年度日本数学会秋季総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-21
• [Presentation] Radially symmetric infinite-time blow up solutions of some semilinear parabolic equations involving critical Sobolev exponent (2007)
  • Author(s): Michinori Ishiwata
  • Organizer: 2007年度日本数学会春季総合分科会
  • Place of Presentation: 埼玉大学
  • Year and Date: 2007-03-29
• [Presentation] The asymptotic behavior of solutions of semilinear parabolic equation with critical Sobolev exponent
  • Author(s): Michinori Ishiwata
  • Organizer: 4th Polish - Japanese Days Current advances in applied nonlinear analysis and mathematical modelling issues
  • Place of Presentation: Warsaw/Madralin, Poland
• [Presentation] On the threshold solutions for semilinear parabolic problems involving critical Sobolev exponent
  • Author(s): Michinori Ishiwata
  • Organizer: Third Euro-Japanese workshop on blow-up, 2008
  • Place of Presentation: Tohoku University, Sendai
• [Presentation] On the concentration phenomena of solutions for semilinear parabolic equations involving critical Sobolev exponent
  • Author(s): Michinori Ishiwata
  • Organizer: Ryukoku Workshop 2008, Recent Progress on Pattern Formation and Dynamics in Mathematical Sciences
  • Place of Presentation: Ryukoku university, Kyoto
• [Presentation] Asymptotic behavior of time-global solutions for a semilinear parabolic equation involving critical Sobolev exponent
  • Author(s): Michinori Ishiwata
  • Organizer: First Chile-Japan workshop on Nonlinear Elliptic and Parabolic PDE, Centro de modelamiento mathematico
  • Place of Presentation: departmento de ingenieria matematica, universidad de Chile, Santiago, Chile
• [Book] これからの非線型偏微分方程式 第10章(「スケール変換不変なエネルギー汎函数をもつ非線型放物型方程式の解の漸近挙動」,共著) (2007)
  • Author(s): 石渡通徳
  • Publisher: 日本評論社
• [Book] これからの非線型偏微分方程式 (2007)
  • Author(s): 石渡通徳
  • Publisher: 日本評論社
• [Book] これからの非線型偏微分方程式(第十章分担執筆) (2007)
  • Author(s): 石渡通徳
  • Total Pages: 303
  • Publisher: 日本評論社