Shape of the solutions and asymptotic analysis for the diffusive equations

• Project/Area Number: 19340036
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Tohoku University
• Principal Investigator: ISHIGE Kazuhiro 東北大学, 大学院・理学研究科, 教授 (90272020)
• Co-Investigator(Kenkyū-buntansha): OGAWA Takayoshi 東北大学, 大学院・理学研究科, 教授 (20224107) ; YANAGIDA Eiji 東京工業大学, 大学院・理工学研究科, 教授 (80174548)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥17,940,000 (Direct Cost: ¥13,800,000、Indirect Cost: ¥4,140,000); Fiscal Year 2010: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2009: ¥5,850,000 (Direct Cost: ¥4,500,000、Indirect Cost: ¥1,350,000); Fiscal Year 2008: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2007: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
• Keywords: 熱方程式 / 最大点挙動 / 等高面 / 大域的挙動 / 爆発問題 / 解の形 / 漸近展開 / 解の凸性 / 解の形状 / 漸近挙動 / 符号変化する解 / 解の最大点挙動 / 解の微分係数の減衰

Research Abstract

We study the shapes of the solutions of diffusive equations, and reveal the relationship between the movement of hot spots and the harmonic functions. Furthermore we studied the asymptotics of the solutions for the semilinear heat equations. In particular, we characterized the location of blow-up set for blow-up solutions, and also gave the decay estimates of the deference between global in time solutions and their asymptotics.

Research Products

• [Journal Article] On a new kind of convexity for solutions of parabolic problems (2011)
  • Author(s): Kazuhiro Ishige and Paolo Salani
  • Journal Title: Discrete Contin. Dyn. Syst. Ser. S Volume: 4巻 Pages: 851-864
  • Peer Reviewed
• [Journal Article] Hot spots for the two dimensional heat equation with a rapidly decaying negative potential (2011)
  • Author(s): Kazuhiro Ishige and Yoshitsugu Kabeya
  • Journal Title: Discrete Contin. Dyn. Syst. Ser. S Volume: 4巻 Pages: 833-849
  • Peer Reviewed
• [Journal Article] Parabolic quasi-concavity for solutions to parabolic problems in convex rings (2010)
  • Author(s): Kazuhiro Ishige and Paolo Salani
  • Journal Title: Math. Nachr. Volume: 283巻 Pages: 1526-1548
  • Peer Reviewed
• [Journal Article] Global solutions of the heat equation with a nonlinear boundary condition (2010)
  • Author(s): Kazuhiro Ishig and Tatsuki Kawakami
  • Journal Title: Calc. Var. Partial Differential Equations Volume: 39巻 Pages: 429-457
  • Peer Reviewed
• [Journal Article] Blow-up set for a semilinear heat equation with small diffusion (2010)
  • Author(s): Yohei Fujishima and Kazuhiro Ishige
  • Journal Title: J. Differential Equations Volume: 249巻 Pages: 1056-1077
  • Peer Reviewed
• [Journal Article] Convexity breaking of the free boundary for porous medium equations (2010)
  • Author(s): Kazuhiro Ishige and Paolo Salani
  • Journal Title: Interfaces Free Bound. Volume: 12巻 Pages: 75-84
  • Peer Reviewed
• [Journal Article] 熱方程式の解の最大点挙動について (2010)
  • Author(s): 石毛和弘
  • Journal Title: 応用数理 Volume: 20巻 Pages: 22-36
  • NAID: 110007580761
  • Peer Reviewed
• [Journal Article] Blow-up set for a semilinear heat equation with small diffusion (2010)
  • Author(s): Yohei Fujishima, Kazuhiro Ishige
  • Journal Title: Journal of Differential Equations Volume: 249 Pages: 1056-1077
  • Peer Reviewed
• [Journal Article] Global solutions of the heat equation with a nonlinear boundary condition (2010)
  • Author(s): Kazuhiro Ishige, Tatsuki Kawakami
  • Journal Title: Calculus of Variations and Partial Differential Equations Volume: 39 Pages: 429-457
  • Peer Reviewed
• [Journal Article] Parabolic quasi-concavity for solutions to parabolic problems in convex rings (2010)
  • Author(s): Kazuhiro Ishige, Paolo Salani
  • Journal Title: Mathematische Nachrichten Volume: 283 Pages: 1526-1548
  • Peer Reviewed
• [Journal Article] Convexity breaking of the free boundary for porous medium equations (2010)
  • Author(s): Kazuhiro Ishige, Paolo Salani
  • Journal Title: Interfaces Free Bound 12巻(印刷中)
  • Peer Reviewed
• [Journal Article] The decay of the solutions for the heat equation with a potential (2009)
  • Author(s): Kazuhiro Ishige, Michinori Ishiwata, and Tatsuki Kawakami
  • Journal Title: Indiana Univ. Math. J. Volume: 58巻 Pages: 2673-2708
  • Peer Reviewed
• [Journal Article] Hot spots for the heat equation with a rapidly decaying negative potential (2009)
  • Author(s): Kazuhiro Ishige and Yoshitsugu Kabeya
  • Journal Title: Adv. Differential Equations Volume: 14巻 Pages: 643-662
  • Peer Reviewed
• [Journal Article] Gradient estimates for the heat equation in the exterior domains under the Neumann boundary condition (2009)
  • Author(s): Kazuhiro Ishige
  • Journal Title: Differential Integral Equations Volume: 22巻 Pages: 401-410
  • 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 Univ.Math.J. 58巻
  • Peer Reviewed
• [Journal Article] (2009)
  • Author(s): Kazuhiro Ishige, Yoshitsugu Kabeya
  • Journal Title: Adv.Differential Equations 14巻
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of solutions for some semilinear heat equations in R^N (2009)
  • Author(s): Kazuhiro Ishige, Tatsuki Kawakami
  • Journal Title: Communications on Pure and Applied Mathematics 8 Pages: 1351-1371
  • Peer Reviewed
• [Journal Article] Hot spots for the heat equation with a rapidly decaying negative potential (2009)
  • Author(s): Kazuhiro Ishige, Yoshitsugu Kabeya
  • Journal Title: Advances in Differential Equations 14 Pages: 643-662
  • 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 Pages: 2673-2708
  • Peer Reviewed
• [Journal Article] Is quasi-concavity preserved by heat flow (2008)
  • Author(s): Kazuhiro Ishige and Paolo Salani
  • Journal Title: Arch. Math. Volume: 90巻 Pages: 450-460
  • Peer Reviewed
• [Journal Article] On the Fujita exponent for a semilinear heat equation with a potential term (2008)
  • Author(s): Kazuhiro Ishige
  • Journal Title: J. Math. Anal. Appl. Volume: 344巻 Pages: 231-237
  • Peer Reviewed
• [Journal Article] Large time behaviors of hot spots for the heat equation with a potential (2008)
  • Author(s): Kazuhiro Ishige and Yoshitsugu Kabeya
  • Journal Title: J. Differential Equations Volume: 244巻 Pages: 2934-2962
  • Peer Reviewed
• [Journal Article] Is quasi-concavity preserved by heat flow? (2008)
  • Author(s): Kazuhiro Ishige, Paolo Salani
  • Journal Title: Arch.Math. 90巻
  • Peer Reviewed
• [Journal Article] Large time behaviors of hot spots for the heat equation with a potential (2008)
  • Author(s): Kazuhiro Ishige
  • Journal Title: Journal of Differential Equations 244 Pages: 2934-2962
  • Peer Reviewed
• [Journal Article] On the Fujita exponent for a semilinear heat equation with a potential term (2008)
  • Author(s): Kazuhiro Ishige
  • Journal Title: Journal of Mathematical Analysis and Applications 344 Pages: 231-237
  • Peer Reviewed
• [Journal Article] Is quasi-concavity preserved by heat. Flow (2008)
  • Author(s): Kazuhiro Ishige
  • Journal Title: Archiv der Mathematik 90 Pages: 450-460
  • Peer Reviewed
• [Journal Article] Slow convergence to zero for a parabolic equation with a supercritical nonlinearity (2008)
  • Author(s): Marek Fila, Michael Winkler, Eiji Yanagida
  • Journal Title: Mathematische Annalen 340 Pages: 477-496
  • Peer Reviewed
• [Journal Article] On the decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball (2007)
  • Author(s): Kazuhiro Ishige and Yoshitsugu Kabeya
  • Journal Title: J. Math. Soc. Japan Volume: 59巻 Pages: 861-898
  • NAID: 10019540137
  • Peer Reviewed
• [Journal Article] Large time behaviors of hot spots for the heat equation with a potential (2007)
  • Author(s): Kazuhiro Ishige, Yoshitsugu Kabeya
  • Journal Title: J.Differential Equations 244巻
  • Peer Reviewed
• [Journal Article] On the decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball (2007)
  • Author(s): Kazuhiro Ishige, Yoshitsugu Kabeya
  • Journal Title: Journal of Mathematical Society of Japan 59 Pages: 861-898
  • NAID: 10019540137
  • Peer Reviewed
• [Journal Article] Movement of hot spots on the exterior domain of a ball under the Dirichlet boundary condition (2007)
  • Author(s): Kazuhiro Ishige
  • Journal Title: Advances in Differential Equations 12 Pages: 1135-1166
  • Peer Reviewed
• [Presentation] Convexity breaking of the free boundary for porous medium equations (2011)
  • Author(s): 石毛和弘
  • Organizer: The 3rd Scienceweb GCOE International Symposium
  • Place of Presentation: 東北大学
  • Year and Date: 2011-02-17
• [Presentation] Blow-up for a semilinear parabolic equation with large diffusion on R^N (2010)
  • Author(s): 石毛和弘
  • Organizer: 4th Euro-Japanese Workshop on Blow-up
  • Place of Presentation: ライデン
  • Year and Date: 2010-09-10
• [Presentation] Global solutions for a semilinear heat equation in the exterior domain of a compact set (2010)
  • Author(s): 石毛和弘
  • Organizer: RIMS研究集会「保存則と幾何学的偏微分方程式およびその応用」
  • Place of Presentation: 京都大学数理科学研究所
  • Year and Date: 2010-06-09
• [Presentation] Blow-up for a semilinear parabolic equation with large diffusion on R^N (2009)
  • Author(s): 石毛和弘
  • Organizer: The second Chile-Japan workshop on nonlinear elliptic and parabolic PDEs
  • Place of Presentation: 明治大学
  • Year and Date: 2009-12-02
• [Presentation] Hot spots for the heat equation with a rapidly decaying negative potential (2009)
  • Author(s): 石毛和弘
  • Organizer: 1st Italian-Japanese workshop on geometric properties for parabolic and elliptic PDE's
  • Place of Presentation: 東北大学
  • Year and Date: 2009-06-16
• [Presentation] Movement of hot spots and blow-up problem for a semilinear heat equation (2008)
  • Author(s): 石毛和弘
  • Organizer: Equadiff 07 minisymposium 「Qualitive theory of elliptic and parabolic PDE」
  • Place of Presentation: ウィーン
  • Year and Date: 2008-08-07
• [Presentation] Decay rate of the derivatives of the solutions of the heat equations in the exterior domain of a ball (2007)
  • Author(s): 石毛和弘
  • Organizer: 研究集会「微分方程式についての最近の話題」
  • Place of Presentation: 群馬大学東京オフィス
  • Year and Date: 2007-10-21