Studies on the qualitative theory and singularities of nonlinear partial differential equations

• Project/Area Number: 16H02151
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Meiji University (2017-2020); The University of Tokyo (2016)
• Principal Investigator: Matano Hiroshi 明治大学, 研究・知財戦略機構(中野), 特任教授 (40126165)
• Co-Investigator(Kenkyū-buntansha): 宮本 安人 東京大学, 大学院数理科学研究科, 准教授 (90374743)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥27,300,000 (Direct Cost: ¥21,000,000、Indirect Cost: ¥6,300,000); Fiscal Year 2019: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2018: ¥5,590,000 (Direct Cost: ¥4,300,000、Indirect Cost: ¥1,290,000); Fiscal Year 2017: ¥7,930,000 (Direct Cost: ¥6,100,000、Indirect Cost: ¥1,830,000); Fiscal Year 2016: ¥8,710,000 (Direct Cost: ¥6,700,000、Indirect Cost: ¥2,010,000)
• Keywords: 非線形偏微分方程式 / 定性的理論 / 進行波 / 界面運動 / 解の爆発 / 特異極限 / 自由境界問題 / 力学系 / 解析学 / 非線形偏微分方程式論 / 反応拡散系 / 解の特異性 / 順序保存力学系 / 非線形拡散方程式 / 広がり波面 / 関数方程式論 / 無限次元力学系 / 疫病伝播モデル

Outline of Final Research Achievements

We studied qualitative properties of solutions of nonlinear diffusion equations, with emphasis on the front propagation phenomena. Here are the main achievements.

(1) We succeeded in determining the spreading speed of fronts in predator-prey reaction-diffusion models. (2) We studied disease-spreading models on spatially inhomogeneous media and revealed the directional dependence of the spreading speed. (3) We showed that the shape of the spreading front of an anisotropic diffusion equation approaches the Wulff shape associated with the anisotropy of the diffusion. (4) We classified the dynamics of solutions of semilinear diffusion equations on R. Interesting results have also been obtained for equations with time-periodic or spatially periodic nonlinearities. (5) We considered the singular-limit of volume-preserving Allen-Cahn equations and studies in detail the initial formation of interfaces.

Academic Significance and Societal Importance of the Research Achievements

非線形拡散方程式に現れる進行波や波面の広がり現象は，数理生態学や物理学など応用上の観点からも重要であり，近年盛んに研究が行われている．しかし比較定理が成り立たない連立系の場合は，多くが未解明である．本研究では，捕食者被食者系（predator-preyモデル）における波面の広がり現象を研究し，波面の広がりが，確定した速度で起こることを初めて明らかにした．これは，反応拡散系における波面の伝播現象の研究に一つの突破口を開くものである．また，空間１次元半線形拡散方程式のダイナミクスの研究では，交点数非増大則を巧妙に用いた新しい論法を確立した．この手法は，今後，他の研究にも大きく役立つと思われる．

Research Products

• [Int'l Joint Research] University of Lorraine/University of Le Havre/University of Paris-sud(フランス)
• [Int'l Joint Research] 華南師範大学(中国)
• [Int'l Joint Research] ペンシルベニア大学(米国)
• [Int'l Joint Research] バース大学(英国)
• [Int'l Joint Research] University of Lorraine/University of Bordeaux/University of Paris-sud(フランス)
• [Int'l Joint Research] University of Crete(ギリシャ)
• [Int'l Joint Research] University of Minnesota(米国)
• [Int'l Joint Research] University of New England(オーストラリア)
• [Int'l Joint Research] University of Warwick(英国)
• [Int'l Joint Research] University of Cergy-Pontoise/University of Paris-Sud/University of Montpellier(フランス)
• [Int'l Joint Research] Free University of Berlin(ドイツ)
• [Int'l Joint Research] University of Minnesota(米国)
• [Int'l Joint Research] University of Crete(ギリシャ)
• [Int'l Joint Research] University of New England(オーストラリア)
• [Int'l Joint Research]
• [Int'l Joint Research] ミネソタ大学(米国)
• [Int'l Joint Research] ボルドー大学/フランス高等社会科学研究院/ロレーヌ大学(フランス)
• [Int'l Joint Research] クレタ大学(ギリシャ)
• [Journal Article] Generation of Interface for Solutions of the Mass Conserved Allen--Cahn Equation (2020)
  • Author(s): Hilhorst Danielle、Matano Hiroshi、Nguyen Thanh Nam、Weber Hendrik
  • Journal Title: SIAM Journal on Mathematical Analysis Volume: 52 Issue: 3 Pages: 2624-2654
  • DOI: 10.1137/18m1204747
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Dynamics of Time-Periodic Reaction-Diffusion Equations with Front-Like Initial Data on $\mathbb{R}$ (2020)
  • Author(s): Ding Weiwei、Matano Hiroshi
  • Journal Title: SIAM Journal on Mathematical Analysis Volume: 52 Issue: 3 Pages: 2411-2462
  • DOI: 10.1137/19m1268987
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Convergence and structure theorems for order-preserving dynamical systems with mass conservation (2020)
  • Author(s): Ogiwara Toshiko、Hilhorst Danielle、Matano Hiroshi
  • Journal Title: Discrete & Continuous Dynamical Systems - A Volume: 40 Issue: 6 Pages: 3883-3907
  • DOI: 10.3934/dcds.2020129
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Existence and uniqueness of propagating terraces (2019)
  • Author(s): Giletti Thomas、Matano Hiroshi
  • Journal Title: Communications in Contemporary Mathematics Volume: 22 Issue: 06 Pages: 1950055-1950055
  • DOI: 10.1142/s021919971950055x
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Dynamics of time-periodic reaction-diffusion equations with compact initial support on R (2019)
  • Author(s): Ding Weiwei、Matano Hiroshi
  • Journal Title: Journal de Mathematiques Pures et Appliquees Volume: 131 Pages: 326-371
  • DOI: 10.1016/j.matpur.2019.09.010
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Spreading speeds for multidimensional reaction-diffusion systems of the prey?predator type (2019)
  • Author(s): Ducrot Arnaud、Giletti Thomas、Matano Hiroshi
  • Journal Title: Calculus of Variations and Partial Differential Equations Volume: 58 Issue: 4
  • DOI: 10.1007/s00526-019-1576-2
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Asymptotic behavior of spreading fronts in the anisotropic Allen-Cahn equation on R^n (2019)
  • Author(s): Hiroshi Matano, Yoichiro Mori and Mitsunori Nara
  • Journal Title: A.I.H. Analyse non Lineaire Volume: 36 Issue: 3 Pages: 585-626
  • DOI: 10.1016/j.anihpc.2018.07.003
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Existence and uniqueness of propagating terraces (2019)
  • Author(s): Thomas Giletti and Hiroshi Matano
  • Journal Title: Communications in Contemporary Mathatematics Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type (2019)
  • Author(s): Arnaud Ducrot, Thomas Giletti and Hiroshi Matano
  • Journal Title: Calculus of Variations and PDEs Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A limit equation and bifurcation diagrams for semilinear elliptic equations with general supercritical growth (2018)
  • Author(s): Yasuhito Miyamoto
  • Journal Title: Journal of Differential Equations Volume: 264 Issue: 4 Pages: 2684-2707
  • DOI: 10.1016/j.jde.2017.10.034
  • Peer Reviewed
• [Journal Article] Non-uniqueness of Solutions of a Semilinear Heat Equation with Singular Initial Data (2017)
  • Author(s): Fila Marek、Matano Hiroshi、Yanagida Eiji
  • Journal Title: Gurevich P., Hell J., Sandstede B., Scheel A. (eds) Patterns of Dynamics Volume: - Pages: 138-148
  • DOI: 10.1007/978-3-319-64173-7_9
  • ISBN: 9783319641720, 9783319641737
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Monomolecular reaction networks: Flux-influenced sets and balloons (2017)
  • Author(s): Vassena Nicola、Matano Hiroshi
  • Journal Title: Mathematical Methods in the Applied Sciences Volume: 40 Issue: 18 Pages: 7722-7736
  • DOI: 10.1002/mma.4557
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] An entire solution of a bistable parabolic equation on R with two colliding pulses (2017)
  • Author(s): H. Matano and P. Polacik
  • Journal Title: J. Functional Analysis Volume: 272 Issue: 5 Pages: 1956-1979
  • DOI: 10.1016/j.jfa.2016.11.006
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] ynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences (2016)
  • Author(s): H. Matano and P. Polacik
  • Journal Title: Comm. Partial Differential Equations Volume: 41 Issue: 5 Pages: 785-811
  • DOI: 10.1080/03605302.2016.1156697
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Plant disease propagation in a striped periodic medium (2016)
  • Author(s): A. Ducrot and H. Matano
  • Journal Title: Springer Proc. Math. & Stat. (J.M. Cushing et al. (eds.)) Volume: 186 Pages: 121-164
  • DOI: 10.1007/978-81-322-3640-5_8
  • ISBN: 9788132236382, 9788132236405
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Propagation of bistable fronts through a perforated wall (2020)
  • Author(s): Hiroshi Matano
  • Organizer: ハルビン工業大学創立１００周年記念サマースクール(オンライン)，中国
  • Int'l Joint Research / Invited
• [Presentation] Existence of a multi-solition solution for the critical nonlinear heat equation (2019)
  • Author(s): Hiroshi Matano
  • Organizer: Singular Problems, Blow-up, and Regimes with Peaking in Nonlinear PDEs，ロシア民族友好大学, ロシア
  • Int'l Joint Research / Invited
• [Presentation] Propagation of bistable fronts through a perforated wall (2019)
  • Author(s): Hiroshi Matano
  • Organizer: Calculus of Variations: A celebration of Paul Rabinowitz’s 80th birthday，韓国
  • Int'l Joint Research / Invited
• [Presentation] Front propagation in an epidemiological model with mutations II (2019)
  • Author(s): Hiroshi Matano
  • Organizer: Mathematical Analysis for Biology and Ecology，ロレーヌ大学，フランス
  • Int'l Joint Research / Invited
• [Presentation] Propagation of bistable fronts through a perforated wall: Part 1 (2019)
  • Author(s): Hiroshi Matano
  • Organizer: Nonlinear Diffusion Problems，ローマ・サピエンツァ大学, イタリア
  • Int'l Joint Research / Invited
• [Presentation] The basic corrosion model revisited (2019)
  • Author(s): Hiroshi Matano
  • Organizer: Corrosion et Analyse Numeique，パリ南大学，フランス
  • Int'l Joint Research / Invited
• [Presentation] Propagation of bistable fronts through a perforated wall (2019)
  • Author(s): Hiroshi Matano
  • Organizer: New Perspectives in Nonlinear PDE, イスラエル工科大学, イスラエル
• [Presentation] Propagation of bistable fronts through a perforated wall (2019)
  • Author(s): Hiroshi Matano
  • Organizer: 2019 Workshop on Nonlinear Reaction Diffusion Equations，陝西師範大学, 中国
  • Int'l Joint Research / Invited
• [Presentation] Propagation of bistable fronts through a perforated wall (2018)
  • Author(s): Hiroshi Matano
  • Organizer: 国際研究集会"Reaction-Diffusion Equations, Modelling and Social Sciences"（フランス社会科学高等研究院）
  • Int'l Joint Research / Invited
• [Presentation] Front propagation in the presence of obstacles (2018)
  • Author(s): Hiroshi Matano
  • Organizer: 国際研究会議”Recent trends on nonlinear PDEs of elliptic and parabolic type”（メルボルン大学MATRIX研究所）
  • Int'l Joint Research / Invited
• [Presentation] Front propagation in an epidemiological model with mutations (2018)
  • Author(s): Hiroshi Matano
  • Organizer: GDRI ReDiNet 国際研究集会“Recent Progresses in Mathematical Theories in Biological Phenomena”（韓国）
  • Int'l Joint Research / Invited
• [Presentation] Propagation of bistable fronts through a perforated wall (2018)
  • Author(s): Hiroshi Matano
  • Organizer: 国際研究集会"Interacting Particle Systems and Parabolic PDEs"（バンフ国際研究センター）
  • Int'l Joint Research / Invited
• [Presentation] Front propagation in an epidemiological model with mutations (2018)
  • Author(s): Hiroshi Matano
  • Organizer: 第43回偏微分方程式札幌シンポジウム（北海道大学）
  • Int'l Joint Research / Invited
• [Presentation] Existence of a multi-soliton solution for the critical nonlinear heat equation (2018)
  • Author(s): Hiroshi Matano
  • Organizer: 8th Euro-Japanese Workshop on Blow-up （東北大学）
  • Int'l Joint Research / Invited
• [Presentation] Intersection number and applications for semilinear elliptic equations with general supercritical growth (2018)
  • Author(s): Yasuhito Miyamoto
  • Organizer: 8th Euro-Japanese Workshop on Blow-up （東北大学）
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic soliton resolution for the critical nonlinear heat equation (2017)
  • Author(s): Hiroshi Matano
  • Organizer: International Conference on PDEs, Geometric Analysis and Functional Inequalities
  • Place of Presentation: シドニー大学（Sydney, オーストラリア）
  • Year and Date: 2017-03-10
  • Int'l Joint Research / Invited
• [Presentation] Dynamics of one-dimensional reaction-diffusion equations and the extended omega limit set (2017)
  • Author(s): Hiroshi Matano
  • Organizer: ReaDi closing conference: "Reaction-diffusion, propagation and modelling"
  • Int'l Joint Research / Invited
• [Presentation] Front propagation in an epidemiological model with mutations (2017)
  • Author(s): Hiroshi Matano
  • Organizer: Workshop on Dynamics of Parabolic Euqations and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Front propagation in an epidemiological model with mutations (2017)
  • Author(s): Hiroshi Matano
  • Organizer: RIMS 共同研究「非線形現象と反応拡散方程式」
  • Invited
• [Presentation] Front propagation in an epidemiological model with mutations (2017)
  • Author(s): Hiroshi Matano
  • Organizer: ReaDiNet 2017: International Conference on Mathematical Biology
  • Int'l Joint Research / Invited
• [Presentation] Front propagation in the predator-prey model (2017)
  • Author(s): Hiroshi Matano
  • Organizer: One-day workshop on "Reaction-diffusion Equations and Their Applications to Biology"
  • Int'l Joint Research / Invited
• [Presentation] Generation and propagation of fine transition layers for the stochastic Allen-Cahn equation (2017)
  • Author(s): Hiroshi Matano
  • Organizer: University of New England Mathematics Seminar
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic soliton resolution for the critical nonlinear heat equation (2016)
  • Author(s): Hiroshi Matano
  • Organizer: 7th Euro-Japan Workshop on Blow-up
  • Place of Presentation: 数学研究会議センター（Bedlewo, ポーランド）
  • Year and Date: 2016-09-09
  • Int'l Joint Research / Invited
• [Presentation] Soliton resolution for the critical nonlinear heat equation (2016)
  • Author(s): Hiroshi Matano
  • Organizer: Nonlinear Waves 2016: Summer School at IHES
  • Place of Presentation: 高等研究所（Bure-sur-Yvette, フランス）
  • Year and Date: 2016-07-28
  • Int'l Joint Research / Invited
• [Presentation] Front propagation in predator-prey type reaction-diffusion systems (2016)
  • Author(s): Hiroshi Matano
  • Organizer: Patterns of Dynamics
  • Place of Presentation: ベルリン自由大学（Berlin, ドイツ）
  • Year and Date: 2016-07-25
  • Int'l Joint Research / Invited
• [Presentation] Front propagation in an anisotropic Allen-Cahn equation (2016)
  • Author(s): Hiroshi Matano
  • Organizer: Topics in Applied Nonlinear Analysis: Recent Advances and New Trends
  • Place of Presentation: カーネギーメロン大学（Pittsburgh, 米国）
  • Year and Date: 2016-07-18
  • Int'l Joint Research / Invited
• [Presentation] Dynamics of order-preserving systems with mass conservation (2016)
  • Author(s): Hiroshi Matano
  • Organizer: IMA Special Workshop on "Dynamics and Differential Equations"
  • Place of Presentation: ミネソタ大学（Minneapolis, 米国）
  • Year and Date: 2016-06-22
  • Int'l Joint Research / Invited
• [Presentation] Front propagation in predator-prey type reaction-diffusion systems (2016)
  • Author(s): Hiroshi Matano
  • Organizer: International Workshop on "Nonlinear Partial Differential Equations and Applications"
  • Place of Presentation: 上海ニューヨーク大学（上海，中国）
  • Year and Date: 2016-06-13
  • Int'l Joint Research / Invited
• [Remarks] 明治大学教員データベース
  • URL: https://gyoseki1.mind.meiji.ac.jp/mjuhp/KgApp?resId=S002040
• [Funded Workshop] MIMS/CMMA Mini-workshop "Propagation Phenomena in Reaction-Diffusion Systems" （明治大学） (2019)
• [Funded Workshop] Summer School of Applied Analysis (Kusatsu Seminar) （草津セミナーハウス） (2018)
• [Funded Workshop] Workshop on Reaction-Diffusion Systems and Related Topics (2018)
• [Funded Workshop] Winter School of Applied Analysis (Kusatsu Seminar) (2018)
• [Funded Workshop] ミニワークショップ『反応拡散系における進行波と広がり波面』 (2017)
  • Place of Presentation: 東京大学数理科学研究科（東京）