On asymptotic behavior and interface curve of solution for Keller-Segel system of degenerate type

• Project/Area Number: 21540197
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Osaka City University (2011); Tsuda College (2009-2010)
• Principal Investigator: SUGIYAMA Yoshie 大阪市立大学, 大学院・理学研究科, 教授 (60308210)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2009: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
• Keywords: 退化型 / Keller-Segel系 / 爆発解 / 測度値解 / 有限時間爆発 / 時間大域解 / 退化放物型方程式 / デルタ関数 / 退化型Keller-Segel系 / 界面曲線 / 有限伝播性 / 部分正則性定理

Research Abstract

Measure valued solution : The time global solution has been constructed for Keller-Segel system of semi-linear type by expanding the functional space to the measure valued solution.
Asymptotics : We have proved that the solution of Keller-Segel system behaves like a stationary solution asymptotically as time tends to infinity.
Uniqueness : We have established the functional space which the weak solution becomes unique.
Finite speed propagation : The existence of solution and the extinction phenomena has been proved for Keller-Segel system of singular type.

Research Products

• [Journal Article] Measure valued solutions of the 2D Keller-Segel system (2012)
  • Author(s): S.Luckhaus, Y.Sugiyama, J.J.L.Velazquez
  • Journal Title: Archive for Rational Mechanics and Analysis Volume: (掲載決定であるがページ数と巻号は未定)
  • Peer Reviewed
• [Journal Article] Finite speed of propagation in 1-D degenerate Keller-Segel system (2012)
  • Author(s): Y.Sugiyama
  • Journal Title: Math.Nachr Volume: 285 Pages: 744-757
  • Peer Reviewed
• [Journal Article] Uniqueness theorem of mild solutions to the Keller-Segel system in the scaling invariant space (2012)
  • Author(s): H.Kozono, Y.Sugiyama, T.Wachi
  • Journal Title: J.Differential Equations Volume: 252 Pages: 1213-1228
  • Peer Reviewed
• [Journal Article] Extinction and asymptotic behavior of solutions for the Keller-Segel system of fast diffusion type (2011)
  • Author(s): Y. Sugiyama and Y. Yahagi
  • Journal Title: Adv. Math. Sci. Appl. Volume: 35 Pages: 209-219
  • Peer Reviewed
• [Journal Article] Extinction, decay and blow-up for Keller-Segel system of fast diffusion type (2011)
  • Author(s): Y. Sugiyama and Y. Yahagi
  • Journal Title: J. Differential Equations Volume: 250 Pages: 3047-3087
  • Peer Reviewed
• [Journal Article] Self-similar blow-up with a continuo-us range of values of the aggregated mass for a degenerate Keller-Segel system (2011)
  • Author(s): Y. Sugiyama and J. J. L. Velazquez
  • Journal Title: Advances in Differential Equations Volume: 16 Pages: 85-112
  • Peer Reviewed
• [Journal Article] Extinction, decay and blow-up for Keller-Segel system of fast diffusion type (2011)
  • Author(s): Y.Sugiyama, Y.Yahagi
  • Journal Title: J.Differential Equations Volume: 250 Pages: 3047-3087
  • Peer Reviewed
• [Journal Article] Extinction and asymptotic behavior of solutions for the Keller-Segel system of fast diffusion type (2011)
  • Author(s): Y.Sugiyama, Y.Yahagi
  • Journal Title: Adv.Math.Sci.Appl Volume: 35 Pages: 201-219
  • Peer Reviewed
• [Journal Article] Self-similar blow-up with a continuous range of values of the aggregated mass for a degenerate Keller-Segel system (2011)
  • Author(s): Y.Sugiyama, J.J.L.Velazquez
  • Journal Title: Advances in Differential Equations Volume: 16 Pages: 85-112
  • Peer Reviewed
• [Journal Article] Extinction, decay and blow-up for Keller-Segel system of fast diffusion type (2011)
  • Author(s): 杉山由恵, 矢作由美
  • Journal Title: J.Differential Equations Volume: (掲載確定 印刷中により未定)
  • Peer Reviewed
• [Journal Article] Uniqueness and continuity of solution for the initial data in the scaling invariant class of degenerate Keller-Segel system (2011)
  • Author(s): 杉山由恵, 矢作由美
  • Journal Title: J.Evolution Equations Volume: (掲載確定 印刷中により未定)
  • Peer Reviewed
• [Journal Article] Extinction and asymptotic behavior of solutions for the Keller-Segel system of fast diffusion type (2011)
  • Author(s): 杉山由恵, 矢作由美
  • Journal Title: Advances in Mathematical Sciences and Applications Volume: (掲載確定 印刷中により未定)
  • Peer Reviewed
• [Journal Article] Finite speed of propagation in 1-D degenerate Keller-Segel system (2011)
  • Author(s): 杉山由恵
  • Journal Title: Math.Nachr Volume: (掲載確定 印刷中により未定)
  • Peer Reviewed
• [Journal Article] Self similar blow up with different values of the masses for the Keller Segel system with nonlinear diffusion in the critical case (2011)
  • Author(s): 杉山由恵, J.J.L.Velazquez
  • Journal Title: Advances in Differential Equations Volume: 16 Pages: 85-112
  • Peer Reviewed
• [Journal Article] Uniqueness and continuity of solution for the initial data in the scaling invariant class of degenerate Keller-Segel system (2010)
  • Author(s): Y. Sugiyama and Y. Yahagi
  • Journal Title: J. Evol.Equ. Volume: 11 Pages: 319-337
  • Peer Reviewed
• [Journal Article] Uniqueness and regularity of weak solutions for the1-D degenerate Keller-Segel systems Nonlinear Analysis Series A : Theory (2010)
  • Author(s): Y. Sugiyama
  • Journal Title: Methods & Applications Volume: 73 Pages: 2230-2244
  • Peer Reviewed
• [Journal Article] Aronson-Benilan type estimate and the optimal Holder continuity of weak solutions for the1-D degenerate Keller-Segel systems (2010)
  • Author(s): Y. Sugiyama
  • Journal Title: Revista Mathematica Iberoamericana Volume: 26 Pages: 891-913
  • Peer Reviewed
• [Journal Article] ε-regularity theorem and its application to the blow-up solutions of Keller-Segel systems in higher dimensions (2010)
  • Author(s): Y. Sugiyama
  • Journal Title: J. Math. Anal. Appl. Volume: 364 Pages: 51-70
  • Peer Reviewed
• [Journal Article] Asymptotic profile of blow-up solutions of Keller-Segel systems in super-critical cases (2010)
  • Author(s): Y. Sugiyama
  • Journal Title: Differential Integral Equation Volume: 23 Pages: 601-618
  • Peer Reviewed
• [Journal Article] Blow-up criterion via scaling invariant quantities with effect on coefficient growth in Keller-Segel system (2010)
  • Author(s): Y. Sugiyama
  • Journal Title: Differential Integral Equation Volume: 23 Pages: 619-634
  • Peer Reviewed
• [Journal Article] Non-existence of backward self-similar solutions of the Keller-Segel system in the scaling invariant class (2010)
  • Author(s): H. Kozono and Y. Sugiyama, and R. Takada
  • Journal Title: J. Math. Anal. Appl. Volume: 365 Pages: 60-66
  • Peer Reviewed
• [Journal Article] Strong solutions to the Keller-Segel system with the weak L^<n/2> initial data and its application to the blow-up rate (2010)
  • Author(s): H. Kozono and Y. Sugiyama
  • Journal Title: Math. Nachr. Volume: 283 Pages: 732-751
  • Peer Reviewed
• [Journal Article] Uniqueness and regularity of weak solutions for the 1-$D$ degenerate Keller-Segel systems (2010)
  • Author(s): 杉山由恵
  • Journal Title: Nonlinear Analysis Series A : TMA Volume: 73 Pages: 2230-2244
  • Peer Reviewed
• [Journal Article] Aronson-B￥'{e}nilan type estimate and the optimal H￥"{o}lder continuity of weak solutions for the 1-$D$ degenerate Keller-Segel systems. (2010)
  • Author(s): 杉山由恵
  • Journal Title: Revista Mathematica Iberoamericana Volume: 26 Pages: 891-913
  • Peer Reviewed
• [Journal Article] Asymptotic profile of blow-up solutions of Keller-Segel systems in super-critical cases (2010)
  • Author(s): 杉山由恵
  • Journal Title: Differential Integral Equations Volume: 23 Pages: 601-618
  • Peer Reviewed
• [Journal Article] Non-existence of backward self-similar solutions of the Keller-Segel system in the scaling invariant class (2010)
  • Author(s): 小薗英雄, 杉山由恵, 高田了
  • Journal Title: J.Math.Appl.Anal. Volume: 365 Pages: 60-66
  • Peer Reviewed
• [Journal Article] Blow-up criterion via scaling invariant quantities with effect on coefficient growth in Keller-Segel system (2010)
  • Author(s): 杉山由恵
  • Journal Title: Differential Integral Equations Volume: 23 Pages: 619-634
  • Peer Reviewed
• [Journal Article] Strong solutions to the Keller-Segel system with the weak $L^{\frac{n}{2}}$ initial data and its application to the blow-up rate (2010)
  • Author(s): 小薗英雄, 杉山由恵
  • Journal Title: Math.Nachr Volume: 5 Pages: 732-751
  • Peer Reviewed
• [Journal Article] 走化性方程式-非線形拡散と移流項の相互作用について,日本応用数理学会学会誌「応用数理」特集:移流項をもつ反応拡散系 (2009)
  • Author(s): 杉山由恵
  • Journal Title: 日本応用数理学会
  • Peer Reviewed
• [Journal Article] Onε-regularity theorem and asymptotic behaviors of solutions for Keller-Segel systems (2009)
  • Author(s): Y. Sugiyama
  • Journal Title: SIAM J. Math. Anal. Volume: 41 Pages: 1664-1692
  • Peer Reviewed
• [Journal Article] Partial regularity and its application to the blow-up asymptotics of parabolic systems modelling chemotaxis with porous medium diffusion (2009)
  • Author(s): Y. Sugiyama
  • Journal Title: Adv. Study in Pure Math. Volume: 55 Pages: 137-160
  • Peer Reviewed
• [Journal Article] Interfaces for1-D degenerate Keller-Segel systems (2009)
  • Author(s): Y. Sugiyama
  • Journal Title: J. Evol.Equ. Volume: 9 Pages: 123-142
  • Peer Reviewed
• [Journal Article] Global solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in Lr(Rn) (2009)
  • Author(s): H. Kozono and Y. Sugiyama
  • Journal Title: J. Differential Equations. Volume: 247 Pages: 1-32
  • Peer Reviewed
• [Journal Article] On $\epsilon$-regularity theorem and asymptotic behaviors of solutions for Keller-Segel systems (2009)
  • Author(s): Sugiyama, Yoshie
  • Journal Title: SIAM J.Math.Anal. 41 Pages: 1664-1692
  • Peer Reviewed
• [Journal Article] Partial regularity and its application to the blow-up asymptotics of parabolic systems modelling chemotaxis with porous medium diffusion (2009)
  • Author(s): Sugiyama, Yoshie
  • Journal Title: Adv.Stud.Pure Math. 55 Pages: 137-160
  • Peer Reviewed
• [Journal Article] Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces (2009)
  • Author(s): Kozono, Hideo, Sugiyama, Yoshie
  • Journal Title: J.Differential Equations 247 Pages: 1-32
  • Peer Reviewed
• [Journal Article] Interfaces for 1-D degenerate Keller-Segel systems (2009)
  • Author(s): Sugiyama, Yoshie
  • Journal Title: J.Evol.Equ. 9 Pages: 123-142
  • Peer Reviewed
• [Journal Article] Measure valued solutions of the 2D Keller-Segel system
  • Author(s): S. Luckhaus, Y. Sugiyama and J. J. L. Velazquez
  • Journal Title: Archive for Rational Mechanics and Analysis
  • Peer Reviewed
• [Journal Article] Finite speed of propagation in 1-D degenerate Keller-Segel system
  • Author(s): Y. Sugiyama
  • Journal Title: Math. Nachr
  • Peer Reviewed
• [Journal Article] Uniqueness theorem of mild solutions to the Keller-Segel system in the scaling invariant space
  • Author(s): H. Kozono, Y. Sugiyama and T. Wachi
  • Journal Title: J. Differential Equations
  • Peer Reviewed
• [Presentation] Asymptotic stability of stationary solutions to degenerate Keller-Segel systems (2012)
  • Author(s): Y.Sugiyama, Y.Yahagi
  • Organizer: 研究集会「拡散と移流の数理」
  • Place of Presentation: 九州工業大学(招待講演)
  • Year and Date: 2012-12-10
• [Presentation] Asymptotic stability of stationary solutions to degenerate Keller-Segel systems (2012)
  • Author(s): Y.Sugiyama, Y.Yahagi
  • Organizer: The 4^<th> TIMS-OCAMI Joint International Workshop on Differential Geometry and Geometric Analysis
  • Place of Presentation: 国立台湾大学(招待講演)
  • Year and Date: 2012-03-18
• [Presentation] Asymptotic stability of stationary solutions to degenerate Keller-Segel systems (2012)
  • Author(s): Y.Sugiyama, Y.Yahagi
  • Organizer: 京都駅前セミナー
  • Place of Presentation: 龍谷大学(招待講演)
  • Year and Date: 2012-02-21
• [Presentation] Asymptotic stability of stationary solutions to degenerate Keller-Segel systems (2011)
  • Author(s): Y.Sugiyama, Y.Yahagi
  • Organizer: 日本数学会関数方程式分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-30
• [Presentation] Measure valued solutions of the 2D Keller-Segel system (2011)
  • Author(s): 杉山由恵
  • Organizer: 第13回広島応用解析セミナー
  • Place of Presentation: 広島大学(招待講演)
  • Year and Date: 2011-08-23
• [Presentation] Asymptotic stability of stationary solutions to degenerate Keller-Segel systems「変分問題の展開-発展方程式論における変分的方法」 (2011)
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-06-06
• [Presentation] Asymptotic stability of stationary solutions to degenerate Keller-Segel systems (2011)
  • Author(s): Y.Sugiyama, Y.Yahagi
  • Organizer: 京都大学数理解析研究所
  • Place of Presentation: 京都大学(招待講演)
  • Year and Date: 2011-06-06
• [Presentation] Measure valued solutions of the 2D Keller-Segel system (2011)
  • Author(s): Y.Sugiyama
  • Organizer: 大阪市立大学談話会
  • Place of Presentation: 大阪市立大学(招待講演)
  • Year and Date: 2011-04-20
• [Presentation] Measure valued solutions of the 2D Keller-Segel system (2011)
  • Author(s): S.Luckhaus, Y.Sugiyama, J.J.L.Velazquez
  • Organizer: 京都大学NLPDEセミナー
  • Place of Presentation: 京都大学(招待講演)
  • Year and Date: 2011-04-15
• [Presentation] Measure valued solutions of the 2D Keller-Segel system (2011)
  • Author(s): 杉山由恵
  • Organizer: Seminario
  • Place of Presentation: ICMAT(Spain)(招待講演)
  • Year and Date: 2011-02-28
• [Presentation] Measure valued solutions of the 2D Keller-Segel system (2011)
  • Author(s): 杉山由恵
  • Organizer: 第16回早稲田大学「流体数学セミナー」
  • Place of Presentation: 早稲田大学(招待講演)
  • Year and Date: 2011-01-28
• [Presentation] Measure valued solutions of the 2D Keller-Segel, system (2010)
  • Author(s): 杉山由恵
  • Organizer: 熊本における偏微分方程式研究集会
  • Place of Presentation: 熊本大学(招待講演)
  • Year and Date: 2010-11-26
• [Presentation] Measure valued solutions of the2D Keller. Segel system「非線形の諸問題」 (2010)
  • Place of Presentation: 山口大学
  • Year and Date: 2010-09-17
• [Presentation] Measure valued solutions of the 2D Keller-Segel system (2010)
  • Author(s): 杉山由恵
  • Organizer: 第6回非線型の諸問題
  • Place of Presentation: 山口大学(招待講演)
  • Year and Date: 2010-09-17
• [Presentation] Measure valued solutions of the 2D Keller-Segel system (2010)
  • Author(s): 杉山由恵
  • Organizer: 第48回解析ゼミ
  • Place of Presentation: 埼玉大学(招待講演)
  • Year and Date: 2010-06-28
• [Presentation] Extinction and decay for Keller-Segel systems of fast diffusion type (2009)
  • Organizer: 国際研究集会「Mathematical Analysis on the Navier-Stokes Equations and Related Topics, Past and Future
  • Place of Presentation: 神戸大学
  • Year and Date: 2009-12-08
• [Presentation] Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type (2009)
  • Author(s): 杉山由恵
  • Organizer: 日本数学会 関数方程式論分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-26