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Blow up in a degenerate keller--segel system(Eng.)

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Takahiro Hashira

2月24日(土)に、第153解神楽坂解析セミナーで発表しました。

Science
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Blow-up in a degenerate Keller—Segel system Day : February 24, 2018 (Sat.) at Tokyo Univ. of Science Kagurazaka campus Takahiro Hashira (Tokyo Univ. of Science) 第153回神楽坂解析セミナー This talk is based on the joint work with Prof. Sachiko Ishida (Chiba Univ.) and Prof. Tomomi Yokota (Tokyo Univ. of Science)
Problem and background Sketch of the proof Summary Our result 2/11 Plan of this talk
𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ This system describes a biological phenomenon chemotaxis. 3/11 Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance How do cells behave? The cell slime molds move towards higher concentration of the signal substance. This system was proposed by Keller—Segel (1970)
𝚫𝒖 The diffusion term 3/11 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance cells
𝚫𝒖 The diffusion term 3/11 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance
発展 nonlinear type: 𝛁 ⋅ 𝑫 𝒖, 𝒗 𝛁𝒖 degenerate type: 𝚫𝒖 𝒎 develop etc. 3/11 𝚫𝒖 The diffusion term 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance
発展 nonlinear type: 𝛁 ⋅ 𝑫 𝒖, 𝒗 𝛁𝒖 degenerate type: 𝚫𝒖 𝒎 etc. 3/11 𝚫𝒖 The diffusion term 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance −𝛁 ⋅ 𝒖𝛁𝒗 The chemotaxis term develop
発展 nonlinear type: 𝛁 ⋅ 𝑫 𝒖, 𝒗 𝛁𝒖 degenerate type: 𝚫𝒖 𝒎 etc. 3/11 𝚫𝒖 The diffusion term 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance −𝛁 ⋅ 𝒖𝛁𝒗 The chemotaxis term develop
発展 nonlinear type: 𝛁 ⋅ 𝑫 𝒖, 𝒗 𝛁𝒖 degenerate type: 𝚫𝒖 𝒎 etc. 3/11 𝚫𝒖 The diffusion term 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance −𝛁 ⋅ 𝒖𝛁𝒗 The chemotaxis term develop
3/11 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance v.s. blow-up boundedness ● Herrero—Velazquez (1997), Horstmann—Wang (2001), Winkler (2013) 𝐬𝐮𝐩 𝒙∈ഥ𝛀 |𝒖 𝒙, 𝒕 | → ∞ 𝒕 → ∃𝑻 𝟎 ∃ 𝒖 𝟎, 𝒗 𝟎 : initial data s.t. ● Nagai—Senba—Yoshida (1997), Nagai—Ogawa (2011), Winkler (2010), Cao (2015) ∃ 𝒖 𝟎, 𝒗 𝟎 : initial data ∃𝑪 > 𝟎 s.t. 𝐬𝐮𝐩 𝒙∈ഥ𝛀 ( 𝒖 𝒙, 𝒕 + |𝒗(𝒙, 𝒕)|) ≤ 𝑪 ∀𝒕 ∈ 𝟎, ∞ Aggregation Diffusion
𝒖 𝒕 = ∆𝒖 𝒎 − 𝛁 ∙ 𝒖 𝒒−𝟏 𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝛁𝒖 𝒎 ∙ = 𝛁𝒗 ∙ = 𝟎, 𝒖 𝒙, 𝟎 = 𝒖 𝟎(𝒙), 𝒗 𝒙, 𝟎 = 𝒗 𝟎(𝒙), 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝝏𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 . (KS) In this talk we consider the following degenerate Keller—Segel system: 𝒖 𝟎 ≥ 𝟎 in 𝛀 , 𝒖 𝟎 ∈ 𝑳∞ 𝛀 , 𝛁𝒖 𝟎 𝒎 ∈ 𝑳 𝟐 𝛀 , 𝒗 𝟎 ≥ 𝟎 in 𝛀 , 𝒗 𝟎 ∈ 𝑾 𝟏,∞ 𝛀 . 𝛀 ≔ 𝒙 ∈ ℝ 𝑵 𝒙 < 𝑹} (𝑵 ≥ 𝟐, 𝑹 > 𝟎) 𝒖, 𝒗 : unknown functions 4/11 ν ν Problem 𝒎 ≥ 𝟏, 𝒒 ≥ 𝟐 : constants, ν: the outer normal vector of 𝝏𝛀 𝒖 𝟎 , 𝒗 𝟎 : given functions such that
Finite or infinite time blow-up𝒒 > 𝒎 + 𝟐 𝑵 𝒒 < 𝒎 + 𝟐 𝑵 Tao—Winkler (2012), Ishida—Seki—Yokota (2014) ´ Ishida—Yokota (2013) (KS) : 𝒖 𝒕 = 𝚫𝒖 𝒎 − 𝛁 ⋅ 𝒖 𝒒−𝟏 𝛁𝒗𝒖 𝒖𝒎 𝒒−𝟏 𝒎 : power of diffusion 𝒒 : power of aggregation 5/11 Global existence and boundedness𝒒 < 𝒎 + 𝟐 𝑵 Ishida—Yokota (2012), Ishida—Seki—Yokota (2014) Known result (KS) : 𝒖 𝒕 = 𝚫(𝒖 + 𝜺) 𝒎−𝛁 ⋅ 𝒖 + 𝜺 𝒒−𝟐 𝒖𝛁𝒗 (𝜺 > 𝟎)𝜺 Cieslak—Stinner (2012, 2014) 𝒒 > 𝒎 + 𝟐 𝑵 Finite-time blow-up Global existence and boundedness
Finite or infinite time blow-up𝒒 > 𝒎 + 𝟐 𝑵 Ishida—Yokota (2013) (KS) : 𝒖 𝒕 = 𝚫𝒖 𝒎 − 𝛁 ⋅ 𝒖 𝒒−𝟏 𝛁𝒗𝒖 𝒖𝒎 𝒒−𝟏 𝒎 : power of diffusion 𝒒 : power of aggregation 5/11 Known result (KS) : 𝒖 𝒕 = 𝚫(𝒖 + 𝜺) 𝒎−𝛁 ⋅ 𝒖 + 𝜺 𝒒−𝟐 𝒖𝛁𝒗 (𝜺 > 𝟎)𝜺 Cieslak—Stinner (2012, 2014) 𝒒 > 𝒎 + 𝟐 𝑵 Finite-time blow-up ´ The purpose of this study To build conditions for initial data such that the corresponding solution of (KS) blows up in finite time. There is a gap between two results.
Main theorem Let 𝑵 ≥ 𝟐, 𝒎 ≥ 𝟏, 𝒒 ≥ 𝟐, and assume that 𝑵, 𝒎, 𝒒 satisfy (super-critical condition). For all 𝑴, 𝑨 > 𝟎 there exist constants 𝑲 𝑴, 𝑨 , 𝑻 𝑴, 𝑨 > 𝟎 such that for any 𝒒 > 𝒎 + 𝟐 𝑵 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑩 𝑴, 𝑨 ≔ 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑳∞ 𝛀 × 𝑾 𝟏,∞ 𝛀 𝛁𝒖 𝟎 𝒎 ∈ 𝑳 𝟐 𝛀 , 𝑮 𝒖 𝟎 ∈ 𝑳 𝟏 𝛀 , 𝒖 𝟎, 𝒗 𝟎: radially symmetric and nonnegative, න 𝛀 𝒖 𝟎 = 𝑴 , 𝒗 𝟎 𝑯 𝟏(𝛀) ≤ 𝑨, 𝑭 𝒖 𝟎, 𝒗 𝟎 ≤ −𝑲(𝑴, 𝑨) ቄ ቄ ※ 6/11 𝑭 𝒖 𝟎, 𝒗 𝟎 ≔ 𝟏 𝟐 𝒗 𝟎 𝑯 𝟏 𝛀 𝟐 − න 𝛀 𝒖 𝟎 𝒗 𝟎 + න 𝛀 𝑮 𝒖 𝟎 .𝑮 𝒖 𝟎 ≔ න 𝒔 𝟎 𝒖 𝟎 න 𝒔 𝟎 𝝈 𝒎𝝉 𝒎−𝒒 𝒅𝝉 𝒔𝝈, ⟹ each corresponding weak solution 𝒖, 𝒗 of (KS) blows up at 𝑻 ≤ 𝑻 𝑴, 𝑨 < ∞, i.e., max 𝐥𝐢𝐦 𝐬𝐮𝐩 𝒕→𝑻 𝒖(⋅, 𝒕) 𝑳∞(𝛀) = ∞. max
Main theorem Let 𝑵 ≥ 𝟐, 𝒎 ≥ 𝟏, 𝒒 ≥ 𝟐, and assume that 𝑵, 𝒎, 𝒒 satisfy (super-critical condition). For all 𝑴, 𝑨 > 𝟎 there exist constants 𝑲 𝑴, 𝑨 , 𝑻 𝑴, 𝑨 > 𝟎 such that for any 𝒒 > 𝒎 + 𝟐 𝑵 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑩 𝑴, 𝑨 ≔ 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑳∞ 𝛀 × 𝑾 𝟏,∞ 𝛀 𝛁𝒖 𝟎 𝒎 ∈ 𝑳 𝟐 𝛀 , 𝑮 𝒖 𝟎 ∈ 𝑳 𝟏 𝛀 , 𝒖 𝟎, 𝒗 𝟎: radially symmetric and nonnegative, න 𝛀 𝒖 𝟎 = 𝑴 , 𝒗 𝟎 𝑯 𝟏(𝛀) ≤ 𝑨, 𝑭 𝒖 𝟎, 𝒗 𝟎 ≤ −𝑲(𝑴, 𝑨) ቄ ቄ 6/11 ⟹ each corresponding weak solution 𝒖, 𝒗 of (KS) blows up at 𝑻 ≤ 𝑻 𝑴, 𝑨 < ∞, i.e., max 𝐥𝐢𝐦 𝐬𝐮𝐩 𝒕→𝑻 𝒖(⋅, 𝒕) 𝑳∞(𝛀) = ∞. max aggregation > diffusion ⟹ Finite-time blow-up!!
Dissipation rate of (KS) : Lyapunov function of (KS) : 𝑮 𝒖 ≔ 𝐥𝐢𝐦 𝜹→𝟎 න 𝒔 𝟎 𝒖 න 𝒔 𝟎 𝝈 𝒎𝝉 𝒎−𝟏 𝝉 𝒒−𝟏 + 𝜹 𝒅𝝉𝒅𝝈 𝑫 𝒖, 𝒗 ≔ න 𝛀 𝒗 𝒕 𝟐 + 𝐥𝐢𝐦 𝜹→𝟎 න 𝛀 𝛁𝒖 𝒎 − 𝒖 𝒒−𝟏 𝛁𝒗 𝒖 Τ𝒒−𝟏 𝟐 + 𝜹 𝟐 11/17/11 𝑭 𝒖, 𝒗 ≔ 𝟏 𝟐 න 𝛀 𝛁𝒗 𝟐 + 𝟏 𝟐 න 𝛀 𝒗 𝟐 − න 𝛀 𝒖𝒗 + න 𝛀 𝑮(𝒖) Outline of the proof 𝑭 𝒖 𝒕 , 𝒗 𝒕 ≥ −𝑪 𝑫 𝒖 𝒕 , 𝒗 𝒕 + 𝟏 𝜽 , ∃𝜽 ∈ 𝟎, 𝟏 ∃𝑪 > 𝟎 s.t. 𝒕 ∈ 𝟎, 𝑻 Our goal
11/17/11 𝑭 𝒖, 𝒗 ≔ 𝟏 𝟐 න 𝛀 𝛁𝒗 𝟐 + 𝟏 𝟐 න 𝛀 𝒗 𝟐 − න 𝛀 𝒖𝒗 + න 𝛀 𝑮(𝒖) Outline of the proof 𝑭 𝒖 𝒕 , 𝒗 𝒕 ≥ −𝑪 𝑫 𝒖 𝒕 , 𝒗 𝒕 + 𝟏 𝜽 , ∃𝜽 ∈ 𝟎, 𝟏 ∃𝑪 > 𝟎 s.t. 𝒕 ∈ 𝟎, 𝑻 Our goal ≥ 𝟎 ≥ 𝟎 ≥ 𝟎≤ 𝟎 න 𝛀 𝒖𝒗Estimate 𝑫 𝒖, 𝒗by Lyapunov function of (KS) : 𝑮 𝒖 ≔ 𝐥𝐢𝐦 𝜹→𝟎 න 𝒔 𝟎 𝒖 න 𝒔 𝟎 𝝈 𝒎𝝉 𝒎−𝟏 𝝉 𝒒−𝟏 + 𝜹 𝒅𝝉𝒅𝝈
Step 1 Step 2 Step 3 Step 4 න 𝛀 𝒖𝒗 න 𝛀 𝒖𝒗 න 𝛀 𝛁𝒗 𝟐 Estimate by 𝑫 𝒖, 𝒗 7/11 𝑭 𝒖 𝒕 , 𝒗 𝒕 ≥ −𝑪 𝑫 𝒖 𝒕 , 𝒗 𝒕 + 𝟏 𝜽 , ∃𝜽 ∈ 𝟎, 𝟏 ∃𝑪 > 𝟎 s.t. 𝒕 ∈ 𝟎, 𝑻 Our goal Outline of the proof Estimate by 𝛀 𝒓 𝟎 𝑹 න 𝛀 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 𝑫 𝒖, 𝒗Estimate by න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 𝑫 𝒖, 𝒗Estimate by
න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ −𝑪 𝟏 න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 + 𝜹 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 + 𝑪 𝟐 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝑪 𝟑 𝒗 𝑳 𝟐 𝟐 OK!! Left-hand side 8/11 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. ( 𝒖 > 𝟎 )𝒇 ≔ −𝒓 𝟏−𝑵 𝒓 𝑵−𝟏 𝒗 𝒓 𝒓 + 𝒗 − 𝒖 , 𝒈 ≔ 𝒖 𝒎 𝒓 − 𝒖 𝒒−𝟏 𝒗 𝒓 𝒖( Τ𝒒−𝟏) 𝟐 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ 𝝁 + 𝑪 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝒓 𝟎 𝒈 𝑳 𝟐 𝟐 + 𝒗 𝑳 𝟐 𝟐 + 𝟏න 𝛀 𝑮(𝒖) Key Lemma 𝒓 𝟎 ⋅ 𝑫 𝒖, 𝒗 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. Sketch of the proof
8/11 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. ( 𝒖 > 𝟎 )𝒇 ≔ −𝒓 𝟏−𝑵 𝒓 𝑵−𝟏 𝒗 𝒓 𝒓 + 𝒗 − 𝒖 , 𝒈 ≔ 𝒖 𝒎 𝒓 − 𝒖 𝒒−𝟏 𝒗 𝒓 𝒖( Τ𝒒−𝟏) 𝟐 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ 𝝁 + 𝑪 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝒓 𝟎 𝒈 𝑳 𝟐 𝟐 + 𝒗 𝑳 𝟐 𝟐 + 𝟏න 𝛀 𝑮(𝒖) Key Lemma 𝒓 𝟎 ⋅ 𝑫 𝒖, 𝒗 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. Known method by[ ]Cieslak—Stinner (2012)´ 𝒖 𝒕 = 𝚫 𝒖 + 𝜺 𝒎 − 𝛁 ⋅ 𝒖 𝒒−𝟐 𝒖𝛁𝒗+ 𝜺 Key Point!! න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ −𝑪 𝟏 න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 + 𝜹 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 + 𝑪 𝟐 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝑪 𝟑 𝒗 𝑳 𝟐 𝟐 Sketch of the proof OK!! − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 = − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒎(𝒖 + 𝜺) 𝒎−𝟏 (𝒖 ) 𝒒−𝟐 𝒖 𝒓 + න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒈 𝜺 (𝒖 ) Τ𝒒−𝟐 𝟐+ 𝜺 + 𝜺 Left-hand side
8/11 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. ( 𝒖 > 𝟎 )𝒇 ≔ −𝒓 𝟏−𝑵 𝒓 𝑵−𝟏 𝒗 𝒓 𝒓 + 𝒗 − 𝒖 , 𝒈 ≔ 𝒖 𝒎 𝒓 − 𝒖 𝒒−𝟏 𝒗 𝒓 𝒖( Τ𝒒−𝟏) 𝟐 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ 𝝁 + 𝑪 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝒓 𝟎 𝒈 𝑳 𝟐 𝟐 + 𝒗 𝑳 𝟐 𝟐 + 𝟏න 𝛀 𝑮(𝒖) Key Lemma 𝒓 𝟎 ⋅ 𝑫 𝒖, 𝒗 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. Known method by[ ]Cieslak—Stinner (2012)´ 𝒖 𝒕 = 𝚫 𝒖 + 𝜺 𝒎 − 𝛁 ⋅ 𝒖 𝒒−𝟐 𝒖𝛁𝒗+ 𝜺 Key Point!! න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ −𝑪 𝟏 න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 + 𝜹 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 + 𝑪 𝟐 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝑪 𝟑 𝒗 𝑳 𝟐 𝟐 Sketch of the proof OK!! − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 = − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒎(𝒖 + 𝜺) 𝒎−𝟏 (𝒖 ) 𝒒−𝟐 𝒖 𝒓 + න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒈 𝜺 (𝒖 ) Τ𝒒−𝟐 𝟐+ 𝜺 + 𝜺 ≤ 𝒓 𝟎 𝒖 𝑳 𝟏 𝒈 𝜺 𝑳 𝟐 𝜺 𝒒−𝟐 /𝟐 ൗ𝟏 𝟐 = 𝑴𝒓 𝟎 𝒈 𝜺 𝑳 𝟐 𝜺 𝒒−𝟐 /𝟐 𝒖(𝒕) 𝑳 𝟏 = 𝒖 𝟎 𝑳 𝟏 =: 𝑴 Mass conservation law𝜺 = 𝟎 and 𝒒 > 𝟐 Our problem Left-hand side
8/11 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. ( 𝒖 > 𝟎 )𝒇 ≔ −𝒓 𝟏−𝑵 𝒓 𝑵−𝟏 𝒗 𝒓 𝒓 + 𝒗 − 𝒖 , 𝒈 ≔ 𝒖 𝒎 𝒓 − 𝒖 𝒒−𝟏 𝒗 𝒓 𝒖( Τ𝒒−𝟏) 𝟐 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ 𝝁 + 𝑪 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝒓 𝟎 𝒈 𝑳 𝟐 𝟐 + 𝒗 𝑳 𝟐 𝟐 + 𝟏න 𝛀 𝑮(𝒖) Key Lemma 𝒓 𝟎 ⋅ 𝑫 𝒖, 𝒗 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. Key Point!! න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ −𝑪 𝟏 න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 + 𝜹 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 + 𝑪 𝟐 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝑪 𝟑 𝒗 𝑳 𝟐 𝟐 Sketch of the proof OK!! − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 = − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 − න {𝒖≤𝒔 𝟎} 𝒓 𝑵 𝒖𝒗 𝒓 𝒒 > 𝒎 + 𝟐 𝑵 Key Lemma can be proved Separate the interval!! from the upper boundedness of 𝒖 We can make න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 Left-hand side We can apply the known method
Step 1 Step 2 Step 3 Step 4 න 𝛀 𝒖𝒗 න 𝛀 𝒖𝒗 න 𝛀 𝛁𝒗 𝟐 Estimate by 𝑫 𝒖, 𝒗 7/11 𝑭 𝒖 𝒕 , 𝒗 𝒕 ≥ −𝑪 𝑫 𝒖 𝒕 , 𝒗 𝒕 + 𝟏 𝜽 , ∃𝜽 ∈ 𝟎, 𝟏 ∃𝑪 > 𝟎 s.t. 𝒕 ∈ 𝟎, 𝑻 Our goal Outline of the proof Estimate by න 𝛀 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 𝑫 𝒖, 𝒗Estimate by න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 𝑫 𝒖, 𝒗Estimate by
න 𝟎 𝒕 𝑫 𝒖 𝒔 , 𝒗 𝒔 𝒅𝒔 + 𝑭 𝒖 𝒕 , 𝒗 𝒕 = 𝑭 𝒖 𝟎, 𝒗 𝟎 , 𝒕 ∈ 𝟎, 𝑻 Proposition 10/11 Proof of main theorem 𝑭 𝒖 𝒕 , 𝒗 𝒕 ≥ −𝑪 𝑫 𝒖 𝒕 , 𝒗 𝒕 + 𝟏 𝜽 , ∃𝜽 ∈ 𝟎, 𝟏 ∃𝑪 > 𝟎 s.t. 𝒕 ∈ 𝟎, 𝑻 Our goal 𝚽 𝒕 ≔ න 𝟎 𝒕 − 𝑭 𝒖 𝒔 , 𝒗 𝒔 𝟏 𝜽 𝒅𝒔 − 𝑭 𝒖 𝟎, 𝒗 𝟎𝒅 𝒅𝒕 𝚽 𝒕 ≥ 𝜹 𝟎 𝚽 𝒕 𝟏 𝜽 𝚽 𝒕 → ∞ as 𝒕 → ∃𝑻 𝟎 ∴ → ∞ as 𝒕 → 𝑻 𝟎.𝒖(⋅, 𝒕) 𝑳∞(𝛀) 𝟏 𝜽 > 𝟏 ⟹ 𝚽 blows up in finite time
11/11 𝒖𝒕 = ∆𝒖 𝒎 − 𝛁 ∙ 𝒖 𝒒−𝟏 𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. (KS) ቐ Summary 𝒒 > 𝒎 + 𝟐 𝑵 Our result 𝑵 ≥ 𝟑, 𝒎 ≥ 𝟏, Blow-up behavior 𝒒 = 𝟐 > 𝒎 + 𝟐 𝑵 ⟹ Result 1 𝒖 ∶ radially symmetric Blow-up point is only the origin Result 2 Singularity like the Dirac delta function ⟹ In this talk we considered the following degenerate Keller—Segel system: , 𝒖 satisfies some condition Finite-time blow-up
定義 (weak solution) 𝛀 × 𝟎, 𝑻 で定義された球対称な非負値関数の組 𝒖, 𝒗 で次を 満たすものを(KS)の 𝟎, 𝑻 上の弱解という: 𝒖 ∈ 𝑳∞ 𝟎, 𝑻; 𝑳∞ 𝛀 , 𝒖 𝒎 ∈ 𝑳∞ 𝟎, 𝑻; 𝑯 𝟏 𝛀 , 𝒖 𝒎+𝟏 𝟐 ∈ 𝑯 𝟏 𝟎, 𝒕; 𝑳 𝟐 𝛀 , 𝒗 ∈ 𝑳∞ 𝟎, 𝑻; 𝑾 𝟏,∞ 𝛀 , 𝒗 𝒕 ∈ 𝑳 𝟐 𝟎, 𝑻; 𝑳 𝟐 𝛀 . ∀𝝋 ∈ 𝑳 𝟏 𝟎, 𝑻; 𝑯 𝟏 𝛀 ∩ 𝑾 𝟏,𝟏 𝟎, 𝑻; 𝑳 𝟐 𝛀 with supp𝝋 𝒙 ⊂ 𝟎, 𝑻 ; න 𝟎 𝑻 න 𝛀 (𝛁𝒖 𝒎 ⋅ 𝛁 𝝋 − 𝒖 𝒒−𝟏 𝛁𝒖 ⋅ 𝛁𝝋 − 𝒖𝝋 𝒕) 𝒅𝒙𝒅𝒕 = න 𝛀 𝒖 𝟎 𝒙 𝝋 𝒙, 𝟎 𝒅𝒙 , න 𝟎 𝑻 න 𝛀 (𝛁𝒗 ⋅ 𝛁 𝝋 + 𝒗𝝋 − 𝒖𝝋 − 𝒗𝝋 𝒕) 𝒅𝒙𝒅𝒕 = න 𝛀 𝒗 𝟎 𝒙 𝝋 𝒙, 𝟎 𝒅𝒙 . ∃𝑲 > 𝟎 s.t. 𝟐𝒆−𝟐𝒕 𝒎 + 𝟏 𝟐 න 𝟎 𝒕 න 𝛀 𝝏 𝝏𝒔 𝒖 𝒎+𝟏 𝟐 𝟐 𝒅𝒙𝒅𝒔 + 𝟏 𝟐𝒎 න 𝛀 𝛁𝒖 𝒎 𝒕 𝟐 𝒅𝒙 ≤ 𝑲, a.a. 𝒕 ∈ 𝟎, 𝑻 . for all 𝒕 < 𝑻. ただし 𝑲 は 𝒖 𝟎 𝑳 𝟐, 𝛁𝒖 𝟎 𝒎 𝑳 𝟐, 𝒗 𝟎 𝑾 𝟏,∞, 𝒖 𝑳∞(𝟎,𝑻;𝑳∞(𝛀)), 𝒎, 𝒒, 𝑵, 𝛀 に依存する定数. 6/15
𝒖 𝒕 = 𝛁 ⋅ 𝝓 𝒖 𝛁𝒖 − 𝝍 𝒖 𝛁𝒗 , 𝒗 𝒕 = 𝚫𝒗 − 𝒗 + 𝒖 , 𝒙 ∈ 𝛀, 𝒕 > 𝟎, 𝒙 ∈ 𝛀, 𝒕 > 𝟎. に対するcritical-condition [Winkler (2009)] (E) (E) ∃𝒔 𝟎 > 𝟏 ∃𝜺 ∈ 𝟎, 𝟏 ∃𝑲, 𝒌 > 𝟎 s.t. න 𝒔 𝟎 𝒔 𝝈𝝓 𝝈 𝝍 𝝈 𝒅𝝈 ≤ 𝑲 𝒔 𝐥𝐨𝐠 𝒔 , 𝑵 − 𝟐 − 𝜺 𝑵 න 𝒔 𝟎 𝒔 න 𝒔 𝟎 𝝈 𝝓 𝝉 𝝍 𝝉 𝒅𝝉𝒅𝝈 + 𝑲𝒔, න 𝒔 𝟎 𝒔 න 𝒔 𝟎 𝝈 𝝓 𝝉 𝝍 𝝉 𝒅𝝉𝒅𝝈 ≤ 𝒌𝒔 𝐥𝐨𝐠 𝒔 𝜽 𝒌𝒔 𝟐−𝜶 if 𝑵 = 𝟐, if 𝑵 ≥ 𝟑, if 𝑵 = 𝟐, if 𝑵 ≥ 𝟑, with some 𝜽 ∈ 𝟎, 𝟏 , with some 𝜶 > 𝟐 𝑵 , 補足スライド(一般のKeller—Segel 系について) for all 𝒔 ≥ 𝒔 𝟎. ቐ 非有界な(E)の解を与える 初期値 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑪∞ 𝛀 𝟐 が存在
本研究で得られたこと 有限時刻で爆発する弱解を与える 初期値が存在する. 𝒒 < 𝒎 + 𝟐 𝑵 𝒒 > 𝒎 + 𝟐 𝑵 𝒖 𝒕 = 𝚫𝒖 𝒎 − 𝛁 ⋅ (𝒖 𝒒−𝟏 𝛁𝒗), 𝒗 𝒕 = 𝚫𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀, 𝒕 > 𝟎, 𝒙 ∈ 𝛀, 𝒕 > 𝟎. (KS) 補足スライド(先行研究の詳細1) 𝟐 = 𝒎 + 𝟐 𝑵 , 𝒖 𝟎 𝑳 𝟏 < ∃𝑴 𝒄 𝟐 = 𝒎 + 𝟐 𝑵 , 𝒖 𝟎 𝑳 𝟏 > ∃𝑴 𝒄 Blanchet—Laurençot (2013) (𝛀 = ℝ 𝑵 の場合) (𝛀 = ℝ 𝑵 の場合) Laurençot—Mizoguchi (2017) 時間大域的弱解が存在する. 有限時刻で爆発する 弱解を与える初期値が存在する. 𝒒 = 𝟐 , ቐ 𝒒 = 𝟐 , 𝑵 = 𝟑 or 𝟒, Ishida—Yokota (2012) (𝛀 = ℝ 𝑵 の場合) 𝒒 ≥ 𝒎 + 𝟐 𝑵 , 𝒖 𝟎, ∆𝒗 𝟎 : enough small 時間大域的弱解が存在する. Ishida—Yokota (2012), Ishida—Seki—Yokota (2014) 時間大域的弱解が存在し, 一様に有界.
補足スライド(先行研究の詳細2) Keller—Segel 系の有限時刻爆発についての先行研究 𝒖 𝒕 = 𝚫𝒖 𝒎 − 𝛁 ⋅ (𝒖 𝒒−𝟏 𝛁𝒗) 𝒖 𝒕 = 𝚫𝒖 − 𝛁 ⋅ ( 𝒖 𝛁𝒗) 𝒖 𝒕 = 𝚫 𝒖 + 𝜺 𝒎 − 𝛁 ⋅ ( 𝒖 + 𝜺 𝒒−𝟐 𝒖 𝛁𝒗) 𝟐 = 𝒎 + 𝟐 𝑵 , 𝒖 𝟎 𝑳 𝟏 > ∃𝑴 𝒄 Laurençot—Mizoguchi (2017) 有限時刻で爆発する 弱解を与える初期値が存在する. 𝒒 = 𝟐 , 𝑵 = 𝟑 or 𝟒, 本研究で得られたこと 有限時刻で爆発する弱解を与える 初期値が存在する.𝒒 > 𝒎 + 𝟐 𝑵 Winkler (2013) 𝑵 ≥ 𝟑 Cieslak—Stinner (2012, 2014) 有限時刻爆発解を与える初期値が存在する.𝒒 > 𝒎 + 𝟐 𝑵 ´ 有限時刻爆発解を与える初期値が存在する.
補足スライド(爆発解の挙動について) 結果1, 2 𝑵 ≥ 3, 𝒎 ≥ 𝟏, 𝒒 = 𝟐 とし, 𝒎, 𝒒 は を満たすものとする.また, (𝒖, 𝒗)を有限時刻 𝑻 で爆発する(KS)の弱解とする. 𝒒 = 𝟐 > 𝒎 + 𝟐 𝑵 (i) 𝒖 ∈ 𝑳∞ 𝟎, 𝑻; 𝑳 (𝛀) 𝑵(𝟐 − 𝒎) 𝟐 𝒖 は原点においてのみ爆発する.⟹ , 𝒖, 𝒗 は球対称 (ii) 𝐥𝐢𝐦 𝒕→𝑻 න 𝛀 𝒖 𝒙, 𝒕 𝝍 𝒙 𝒅𝒙 𝑵(𝟐 − 𝒎) 𝟐 が存在するすべての 𝝍 ∈ 𝑪 𝒄 ℝ 𝑵 に対して ⟹ 𝒖 は爆発点 𝒙 𝟎 において 次の意味でデルタ関数的な特異性をもつ: ∃𝑴 𝒙 𝟎 > 𝟎 ∃𝑹 > 𝟎 ∃ 𝒕 𝒏 ⊂ 𝟎, 𝑻 with 𝒕 𝒏 → 𝑻 as 𝒏 → ∞, ∃𝒇 ∈ 𝑳 𝟏 𝑩 𝒙 𝟎, 𝑹 ∩ 𝛀 ∩ 𝑳∞ 𝑨 𝒙 𝟎, 𝒓, 𝑹 ∩ 𝛀 for all 𝒓 ∈ (𝟎, 𝑹) s.t., ※ただし 𝜹 ⋅ は, Diracのデルタ関数とする. 𝒖 𝒕 𝒏 → 𝑴 𝒙 𝟎 𝜹 ⋅ −𝒙 𝟎 + 𝒇 weakly* in 𝑳∞ 𝑩 𝒙 𝟎, 𝑹 . 𝑵(𝟐 − 𝒎) 𝟐 𝒏 → ∞
補足スライド(有限時刻爆発解を与える初期値について) 𝑩 𝑴, 𝑨 ≔ 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑳∞ 𝛀 × 𝑾 𝟏,∞ 𝛀 𝛁𝒖 𝟎 𝒎 ∈ 𝑳 𝟐 𝛀 , 𝑮 𝒖 𝟎 ∈ 𝑳 𝟏 𝛀 , 𝒖 𝟎, 𝒗 𝟎 ∶ 非負の球対称関数,න 𝛀 𝒖 𝟎 = 𝑴 , 𝒗 𝟎 𝑯 𝟏(𝛀) ≤ 𝑨, 𝑭 𝒖 𝟎, 𝒗 𝟎 ≤ −𝑲(𝑴, 𝑨) ቄ ቄ 𝑴 > 𝟎 とする. このとき,次で定められた 𝒖 𝜼, 𝒗 𝜼 に対し,∃𝜼 𝟎 > 𝟎 s.t. 𝒖 𝜼, 𝒗 𝜼 ∈ 𝑩 𝑴, 𝑨 ∀𝜼 ∈ 𝟎, 𝜼 𝟎 : 𝒖 𝜼 𝒙 ≔ 𝒂 𝜼 ⋅ 𝜼 𝜷−𝑵 𝒙 𝟐 + 𝜼 𝟐 − 𝜷 𝟐, 𝒗 𝜼 𝒙 ≔ 𝜼 𝜹−𝜸 𝒙 𝟐 + 𝜼 𝟐 − 𝜹 𝟐, if 𝑵 ≥ 𝟑. if 𝑵 = 𝟐,𝐥𝐨𝐠 𝑹 𝜼 −𝜿 𝐥𝐨𝐠 𝑹 𝟐 𝒙 𝟐 + 𝜼 𝟐 , ൞ ただし, 𝜷 > 𝑵, 𝜸 ∈ 𝟏 − 𝒒 + 𝒎 𝑵, 𝑵 − 𝟐 , 𝜹 > 𝑵 𝟐 , 𝒂 𝜼 ≔ 𝜼 𝑵−𝜷 𝑴 ‫׬‬𝛀 𝒙 𝟐 + 𝜼 𝟐 Τ−𝜷 𝟐 𝒅𝒙 . イメージ
− න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 = − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 − න {𝒖≤𝒔 𝟎} 𝒓 𝑵 𝒖𝒗 𝒓 Separate the interval!! න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ 𝝁 + 𝑪 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝒓 𝟎 𝒈 𝑳 𝟐 𝟐 + 𝒗 𝑳 𝟐 𝟐 + 𝟏න 𝛀 𝑮(𝒖) Key Lemma 𝒓 𝟎 ⋅ 𝑫 𝒖, 𝒗 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. Key Point!! We can apply the known method − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 = − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖 𝒎−𝒒+𝟏 𝒖 𝒓 − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖 𝟑−𝒒 𝟐 𝒈 =: 𝑰 𝟏 + 𝑰 𝟐 𝑰 𝟏 = − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖 𝒎−𝒒+𝟏 𝒖 𝒓 = 𝑵 න 𝒖≥𝒔 𝟎 𝒓 𝑵−𝟏 𝑯 𝒖 − න 𝝏 𝒖≥𝒔 𝟎 𝒓 𝑵 𝑯(𝒖) 𝒅𝑺 Integration by parts 𝑯 𝒖 ≔ න 𝒔 𝟎 𝒖 𝝈 𝒎−𝒒+𝟏 𝒅𝝈where 𝒔 𝟎 𝐎 𝒓 𝟎 𝒓 ① ② ③ Collecting values ①, ② and ③, we see that 𝑰 𝟏 ≤ 𝑵 න 𝒖≥𝒔 𝟎 𝒓 𝑵−𝟏 𝑯 𝒖

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Blow up in a degenerate keller--segel system(Eng.)

  • 1. Blow-up in a degenerate Keller—Segel system Day : February 24, 2018 (Sat.) at Tokyo Univ. of Science Kagurazaka campus Takahiro Hashira (Tokyo Univ. of Science) 第153回神楽坂解析セミナー This talk is based on the joint work with Prof. Sachiko Ishida (Chiba Univ.) and Prof. Tomomi Yokota (Tokyo Univ. of Science)
  • 2. Problem and background Sketch of the proof Summary Our result 2/11 Plan of this talk
  • 3. 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ This system describes a biological phenomenon chemotaxis. 3/11 Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance How do cells behave? The cell slime molds move towards higher concentration of the signal substance. This system was proposed by Keller—Segel (1970)
  • 4. 𝚫𝒖 The diffusion term 3/11 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance cells
  • 5. 𝚫𝒖 The diffusion term 3/11 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance
  • 6. 発展 nonlinear type: 𝛁 ⋅ 𝑫 𝒖, 𝒗 𝛁𝒖 degenerate type: 𝚫𝒖 𝒎 develop etc. 3/11 𝚫𝒖 The diffusion term 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance
  • 7. 発展 nonlinear type: 𝛁 ⋅ 𝑫 𝒖, 𝒗 𝛁𝒖 degenerate type: 𝚫𝒖 𝒎 etc. 3/11 𝚫𝒖 The diffusion term 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance −𝛁 ⋅ 𝒖𝛁𝒗 The chemotaxis term develop
  • 8. 発展 nonlinear type: 𝛁 ⋅ 𝑫 𝒖, 𝒗 𝛁𝒖 degenerate type: 𝚫𝒖 𝒎 etc. 3/11 𝚫𝒖 The diffusion term 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance −𝛁 ⋅ 𝒖𝛁𝒗 The chemotaxis term develop
  • 9. 発展 nonlinear type: 𝛁 ⋅ 𝑫 𝒖, 𝒗 𝛁𝒖 degenerate type: 𝚫𝒖 𝒎 etc. 3/11 𝚫𝒖 The diffusion term 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance −𝛁 ⋅ 𝒖𝛁𝒗 The chemotaxis term develop
  • 10. 3/11 𝒖 𝒕 = ∆𝒖 − 𝛁 ∙ 𝒖𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. ቐ Keller—Segel system Introduction 𝒖 ∶ density of cells 𝒗 ∶ concentration of signal substance v.s. blow-up boundedness ● Herrero—Velazquez (1997), Horstmann—Wang (2001), Winkler (2013) 𝐬𝐮𝐩 𝒙∈ഥ𝛀 |𝒖 𝒙, 𝒕 | → ∞ 𝒕 → ∃𝑻 𝟎 ∃ 𝒖 𝟎, 𝒗 𝟎 : initial data s.t. ● Nagai—Senba—Yoshida (1997), Nagai—Ogawa (2011), Winkler (2010), Cao (2015) ∃ 𝒖 𝟎, 𝒗 𝟎 : initial data ∃𝑪 > 𝟎 s.t. 𝐬𝐮𝐩 𝒙∈ഥ𝛀 ( 𝒖 𝒙, 𝒕 + |𝒗(𝒙, 𝒕)|) ≤ 𝑪 ∀𝒕 ∈ 𝟎, ∞ Aggregation Diffusion
  • 11. 𝒖 𝒕 = ∆𝒖 𝒎 − 𝛁 ∙ 𝒖 𝒒−𝟏 𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝛁𝒖 𝒎 ∙ = 𝛁𝒗 ∙ = 𝟎, 𝒖 𝒙, 𝟎 = 𝒖 𝟎(𝒙), 𝒗 𝒙, 𝟎 = 𝒗 𝟎(𝒙), 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝝏𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 . (KS) In this talk we consider the following degenerate Keller—Segel system: 𝒖 𝟎 ≥ 𝟎 in 𝛀 , 𝒖 𝟎 ∈ 𝑳∞ 𝛀 , 𝛁𝒖 𝟎 𝒎 ∈ 𝑳 𝟐 𝛀 , 𝒗 𝟎 ≥ 𝟎 in 𝛀 , 𝒗 𝟎 ∈ 𝑾 𝟏,∞ 𝛀 . 𝛀 ≔ 𝒙 ∈ ℝ 𝑵 𝒙 < 𝑹} (𝑵 ≥ 𝟐, 𝑹 > 𝟎) 𝒖, 𝒗 : unknown functions 4/11 ν ν Problem 𝒎 ≥ 𝟏, 𝒒 ≥ 𝟐 : constants, ν: the outer normal vector of 𝝏𝛀 𝒖 𝟎 , 𝒗 𝟎 : given functions such that
  • 12. Finite or infinite time blow-up𝒒 > 𝒎 + 𝟐 𝑵 𝒒 < 𝒎 + 𝟐 𝑵 Tao—Winkler (2012), Ishida—Seki—Yokota (2014) ´ Ishida—Yokota (2013) (KS) : 𝒖 𝒕 = 𝚫𝒖 𝒎 − 𝛁 ⋅ 𝒖 𝒒−𝟏 𝛁𝒗𝒖 𝒖𝒎 𝒒−𝟏 𝒎 : power of diffusion 𝒒 : power of aggregation 5/11 Global existence and boundedness𝒒 < 𝒎 + 𝟐 𝑵 Ishida—Yokota (2012), Ishida—Seki—Yokota (2014) Known result (KS) : 𝒖 𝒕 = 𝚫(𝒖 + 𝜺) 𝒎−𝛁 ⋅ 𝒖 + 𝜺 𝒒−𝟐 𝒖𝛁𝒗 (𝜺 > 𝟎)𝜺 Cieslak—Stinner (2012, 2014) 𝒒 > 𝒎 + 𝟐 𝑵 Finite-time blow-up Global existence and boundedness
  • 13. Finite or infinite time blow-up𝒒 > 𝒎 + 𝟐 𝑵 Ishida—Yokota (2013) (KS) : 𝒖 𝒕 = 𝚫𝒖 𝒎 − 𝛁 ⋅ 𝒖 𝒒−𝟏 𝛁𝒗𝒖 𝒖𝒎 𝒒−𝟏 𝒎 : power of diffusion 𝒒 : power of aggregation 5/11 Known result (KS) : 𝒖 𝒕 = 𝚫(𝒖 + 𝜺) 𝒎−𝛁 ⋅ 𝒖 + 𝜺 𝒒−𝟐 𝒖𝛁𝒗 (𝜺 > 𝟎)𝜺 Cieslak—Stinner (2012, 2014) 𝒒 > 𝒎 + 𝟐 𝑵 Finite-time blow-up ´ The purpose of this study To build conditions for initial data such that the corresponding solution of (KS) blows up in finite time. There is a gap between two results.
  • 14. Main theorem Let 𝑵 ≥ 𝟐, 𝒎 ≥ 𝟏, 𝒒 ≥ 𝟐, and assume that 𝑵, 𝒎, 𝒒 satisfy (super-critical condition). For all 𝑴, 𝑨 > 𝟎 there exist constants 𝑲 𝑴, 𝑨 , 𝑻 𝑴, 𝑨 > 𝟎 such that for any 𝒒 > 𝒎 + 𝟐 𝑵 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑩 𝑴, 𝑨 ≔ 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑳∞ 𝛀 × 𝑾 𝟏,∞ 𝛀 𝛁𝒖 𝟎 𝒎 ∈ 𝑳 𝟐 𝛀 , 𝑮 𝒖 𝟎 ∈ 𝑳 𝟏 𝛀 , 𝒖 𝟎, 𝒗 𝟎: radially symmetric and nonnegative, න 𝛀 𝒖 𝟎 = 𝑴 , 𝒗 𝟎 𝑯 𝟏(𝛀) ≤ 𝑨, 𝑭 𝒖 𝟎, 𝒗 𝟎 ≤ −𝑲(𝑴, 𝑨) ቄ ቄ ※ 6/11 𝑭 𝒖 𝟎, 𝒗 𝟎 ≔ 𝟏 𝟐 𝒗 𝟎 𝑯 𝟏 𝛀 𝟐 − න 𝛀 𝒖 𝟎 𝒗 𝟎 + න 𝛀 𝑮 𝒖 𝟎 .𝑮 𝒖 𝟎 ≔ න 𝒔 𝟎 𝒖 𝟎 න 𝒔 𝟎 𝝈 𝒎𝝉 𝒎−𝒒 𝒅𝝉 𝒔𝝈, ⟹ each corresponding weak solution 𝒖, 𝒗 of (KS) blows up at 𝑻 ≤ 𝑻 𝑴, 𝑨 < ∞, i.e., max 𝐥𝐢𝐦 𝐬𝐮𝐩 𝒕→𝑻 𝒖(⋅, 𝒕) 𝑳∞(𝛀) = ∞. max
  • 15. Main theorem Let 𝑵 ≥ 𝟐, 𝒎 ≥ 𝟏, 𝒒 ≥ 𝟐, and assume that 𝑵, 𝒎, 𝒒 satisfy (super-critical condition). For all 𝑴, 𝑨 > 𝟎 there exist constants 𝑲 𝑴, 𝑨 , 𝑻 𝑴, 𝑨 > 𝟎 such that for any 𝒒 > 𝒎 + 𝟐 𝑵 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑩 𝑴, 𝑨 ≔ 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑳∞ 𝛀 × 𝑾 𝟏,∞ 𝛀 𝛁𝒖 𝟎 𝒎 ∈ 𝑳 𝟐 𝛀 , 𝑮 𝒖 𝟎 ∈ 𝑳 𝟏 𝛀 , 𝒖 𝟎, 𝒗 𝟎: radially symmetric and nonnegative, න 𝛀 𝒖 𝟎 = 𝑴 , 𝒗 𝟎 𝑯 𝟏(𝛀) ≤ 𝑨, 𝑭 𝒖 𝟎, 𝒗 𝟎 ≤ −𝑲(𝑴, 𝑨) ቄ ቄ 6/11 ⟹ each corresponding weak solution 𝒖, 𝒗 of (KS) blows up at 𝑻 ≤ 𝑻 𝑴, 𝑨 < ∞, i.e., max 𝐥𝐢𝐦 𝐬𝐮𝐩 𝒕→𝑻 𝒖(⋅, 𝒕) 𝑳∞(𝛀) = ∞. max aggregation > diffusion ⟹ Finite-time blow-up!!
  • 16. Dissipation rate of (KS) : Lyapunov function of (KS) : 𝑮 𝒖 ≔ 𝐥𝐢𝐦 𝜹→𝟎 න 𝒔 𝟎 𝒖 න 𝒔 𝟎 𝝈 𝒎𝝉 𝒎−𝟏 𝝉 𝒒−𝟏 + 𝜹 𝒅𝝉𝒅𝝈 𝑫 𝒖, 𝒗 ≔ න 𝛀 𝒗 𝒕 𝟐 + 𝐥𝐢𝐦 𝜹→𝟎 න 𝛀 𝛁𝒖 𝒎 − 𝒖 𝒒−𝟏 𝛁𝒗 𝒖 Τ𝒒−𝟏 𝟐 + 𝜹 𝟐 11/17/11 𝑭 𝒖, 𝒗 ≔ 𝟏 𝟐 න 𝛀 𝛁𝒗 𝟐 + 𝟏 𝟐 න 𝛀 𝒗 𝟐 − න 𝛀 𝒖𝒗 + න 𝛀 𝑮(𝒖) Outline of the proof 𝑭 𝒖 𝒕 , 𝒗 𝒕 ≥ −𝑪 𝑫 𝒖 𝒕 , 𝒗 𝒕 + 𝟏 𝜽 , ∃𝜽 ∈ 𝟎, 𝟏 ∃𝑪 > 𝟎 s.t. 𝒕 ∈ 𝟎, 𝑻 Our goal
  • 17. 11/17/11 𝑭 𝒖, 𝒗 ≔ 𝟏 𝟐 න 𝛀 𝛁𝒗 𝟐 + 𝟏 𝟐 න 𝛀 𝒗 𝟐 − න 𝛀 𝒖𝒗 + න 𝛀 𝑮(𝒖) Outline of the proof 𝑭 𝒖 𝒕 , 𝒗 𝒕 ≥ −𝑪 𝑫 𝒖 𝒕 , 𝒗 𝒕 + 𝟏 𝜽 , ∃𝜽 ∈ 𝟎, 𝟏 ∃𝑪 > 𝟎 s.t. 𝒕 ∈ 𝟎, 𝑻 Our goal ≥ 𝟎 ≥ 𝟎 ≥ 𝟎≤ 𝟎 න 𝛀 𝒖𝒗Estimate 𝑫 𝒖, 𝒗by Lyapunov function of (KS) : 𝑮 𝒖 ≔ 𝐥𝐢𝐦 𝜹→𝟎 න 𝒔 𝟎 𝒖 න 𝒔 𝟎 𝝈 𝒎𝝉 𝒎−𝟏 𝝉 𝒒−𝟏 + 𝜹 𝒅𝝉𝒅𝝈
  • 18. Step 1 Step 2 Step 3 Step 4 න 𝛀 𝒖𝒗 න 𝛀 𝒖𝒗 න 𝛀 𝛁𝒗 𝟐 Estimate by 𝑫 𝒖, 𝒗 7/11 𝑭 𝒖 𝒕 , 𝒗 𝒕 ≥ −𝑪 𝑫 𝒖 𝒕 , 𝒗 𝒕 + 𝟏 𝜽 , ∃𝜽 ∈ 𝟎, 𝟏 ∃𝑪 > 𝟎 s.t. 𝒕 ∈ 𝟎, 𝑻 Our goal Outline of the proof Estimate by 𝛀 𝒓 𝟎 𝑹 න 𝛀 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 𝑫 𝒖, 𝒗Estimate by න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 𝑫 𝒖, 𝒗Estimate by
  • 19. න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ −𝑪 𝟏 න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 + 𝜹 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 + 𝑪 𝟐 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝑪 𝟑 𝒗 𝑳 𝟐 𝟐 OK!! Left-hand side 8/11 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. ( 𝒖 > 𝟎 )𝒇 ≔ −𝒓 𝟏−𝑵 𝒓 𝑵−𝟏 𝒗 𝒓 𝒓 + 𝒗 − 𝒖 , 𝒈 ≔ 𝒖 𝒎 𝒓 − 𝒖 𝒒−𝟏 𝒗 𝒓 𝒖( Τ𝒒−𝟏) 𝟐 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ 𝝁 + 𝑪 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝒓 𝟎 𝒈 𝑳 𝟐 𝟐 + 𝒗 𝑳 𝟐 𝟐 + 𝟏න 𝛀 𝑮(𝒖) Key Lemma 𝒓 𝟎 ⋅ 𝑫 𝒖, 𝒗 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. Sketch of the proof
  • 20. 8/11 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. ( 𝒖 > 𝟎 )𝒇 ≔ −𝒓 𝟏−𝑵 𝒓 𝑵−𝟏 𝒗 𝒓 𝒓 + 𝒗 − 𝒖 , 𝒈 ≔ 𝒖 𝒎 𝒓 − 𝒖 𝒒−𝟏 𝒗 𝒓 𝒖( Τ𝒒−𝟏) 𝟐 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ 𝝁 + 𝑪 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝒓 𝟎 𝒈 𝑳 𝟐 𝟐 + 𝒗 𝑳 𝟐 𝟐 + 𝟏න 𝛀 𝑮(𝒖) Key Lemma 𝒓 𝟎 ⋅ 𝑫 𝒖, 𝒗 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. Known method by[ ]Cieslak—Stinner (2012)´ 𝒖 𝒕 = 𝚫 𝒖 + 𝜺 𝒎 − 𝛁 ⋅ 𝒖 𝒒−𝟐 𝒖𝛁𝒗+ 𝜺 Key Point!! න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ −𝑪 𝟏 න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 + 𝜹 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 + 𝑪 𝟐 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝑪 𝟑 𝒗 𝑳 𝟐 𝟐 Sketch of the proof OK!! − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 = − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒎(𝒖 + 𝜺) 𝒎−𝟏 (𝒖 ) 𝒒−𝟐 𝒖 𝒓 + න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒈 𝜺 (𝒖 ) Τ𝒒−𝟐 𝟐+ 𝜺 + 𝜺 Left-hand side
  • 21. 8/11 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. ( 𝒖 > 𝟎 )𝒇 ≔ −𝒓 𝟏−𝑵 𝒓 𝑵−𝟏 𝒗 𝒓 𝒓 + 𝒗 − 𝒖 , 𝒈 ≔ 𝒖 𝒎 𝒓 − 𝒖 𝒒−𝟏 𝒗 𝒓 𝒖( Τ𝒒−𝟏) 𝟐 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ 𝝁 + 𝑪 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝒓 𝟎 𝒈 𝑳 𝟐 𝟐 + 𝒗 𝑳 𝟐 𝟐 + 𝟏න 𝛀 𝑮(𝒖) Key Lemma 𝒓 𝟎 ⋅ 𝑫 𝒖, 𝒗 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. Known method by[ ]Cieslak—Stinner (2012)´ 𝒖 𝒕 = 𝚫 𝒖 + 𝜺 𝒎 − 𝛁 ⋅ 𝒖 𝒒−𝟐 𝒖𝛁𝒗+ 𝜺 Key Point!! න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ −𝑪 𝟏 න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 + 𝜹 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 + 𝑪 𝟐 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝑪 𝟑 𝒗 𝑳 𝟐 𝟐 Sketch of the proof OK!! − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 = − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒎(𝒖 + 𝜺) 𝒎−𝟏 (𝒖 ) 𝒒−𝟐 𝒖 𝒓 + න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒈 𝜺 (𝒖 ) Τ𝒒−𝟐 𝟐+ 𝜺 + 𝜺 ≤ 𝒓 𝟎 𝒖 𝑳 𝟏 𝒈 𝜺 𝑳 𝟐 𝜺 𝒒−𝟐 /𝟐 ൗ𝟏 𝟐 = 𝑴𝒓 𝟎 𝒈 𝜺 𝑳 𝟐 𝜺 𝒒−𝟐 /𝟐 𝒖(𝒕) 𝑳 𝟏 = 𝒖 𝟎 𝑳 𝟏 =: 𝑴 Mass conservation law𝜺 = 𝟎 and 𝒒 > 𝟐 Our problem Left-hand side
  • 22. 8/11 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. ( 𝒖 > 𝟎 )𝒇 ≔ −𝒓 𝟏−𝑵 𝒓 𝑵−𝟏 𝒗 𝒓 𝒓 + 𝒗 − 𝒖 , 𝒈 ≔ 𝒖 𝒎 𝒓 − 𝒖 𝒒−𝟏 𝒗 𝒓 𝒖( Τ𝒒−𝟏) 𝟐 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ 𝝁 + 𝑪 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝒓 𝟎 𝒈 𝑳 𝟐 𝟐 + 𝒗 𝑳 𝟐 𝟐 + 𝟏න 𝛀 𝑮(𝒖) Key Lemma 𝒓 𝟎 ⋅ 𝑫 𝒖, 𝒗 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. Key Point!! න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ −𝑪 𝟏 න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 + 𝜹 න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 + 𝑪 𝟐 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝑪 𝟑 𝒗 𝑳 𝟐 𝟐 Sketch of the proof OK!! − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 = − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 − න {𝒖≤𝒔 𝟎} 𝒓 𝑵 𝒖𝒗 𝒓 𝒒 > 𝒎 + 𝟐 𝑵 Key Lemma can be proved Separate the interval!! from the upper boundedness of 𝒖 We can make න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 Left-hand side We can apply the known method
  • 23. Step 1 Step 2 Step 3 Step 4 න 𝛀 𝒖𝒗 න 𝛀 𝒖𝒗 න 𝛀 𝛁𝒗 𝟐 Estimate by 𝑫 𝒖, 𝒗 7/11 𝑭 𝒖 𝒕 , 𝒗 𝒕 ≥ −𝑪 𝑫 𝒖 𝒕 , 𝒗 𝒕 + 𝟏 𝜽 , ∃𝜽 ∈ 𝟎, 𝟏 ∃𝑪 > 𝟎 s.t. 𝒕 ∈ 𝟎, 𝑻 Our goal Outline of the proof Estimate by න 𝛀 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 𝑫 𝒖, 𝒗Estimate by න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 𝑫 𝒖, 𝒗Estimate by
  • 24. න 𝟎 𝒕 𝑫 𝒖 𝒔 , 𝒗 𝒔 𝒅𝒔 + 𝑭 𝒖 𝒕 , 𝒗 𝒕 = 𝑭 𝒖 𝟎, 𝒗 𝟎 , 𝒕 ∈ 𝟎, 𝑻 Proposition 10/11 Proof of main theorem 𝑭 𝒖 𝒕 , 𝒗 𝒕 ≥ −𝑪 𝑫 𝒖 𝒕 , 𝒗 𝒕 + 𝟏 𝜽 , ∃𝜽 ∈ 𝟎, 𝟏 ∃𝑪 > 𝟎 s.t. 𝒕 ∈ 𝟎, 𝑻 Our goal 𝚽 𝒕 ≔ න 𝟎 𝒕 − 𝑭 𝒖 𝒔 , 𝒗 𝒔 𝟏 𝜽 𝒅𝒔 − 𝑭 𝒖 𝟎, 𝒗 𝟎𝒅 𝒅𝒕 𝚽 𝒕 ≥ 𝜹 𝟎 𝚽 𝒕 𝟏 𝜽 𝚽 𝒕 → ∞ as 𝒕 → ∃𝑻 𝟎 ∴ → ∞ as 𝒕 → 𝑻 𝟎.𝒖(⋅, 𝒕) 𝑳∞(𝛀) 𝟏 𝜽 > 𝟏 ⟹ 𝚽 blows up in finite time
  • 25. 11/11 𝒖𝒕 = ∆𝒖 𝒎 − 𝛁 ∙ 𝒖 𝒒−𝟏 𝛁𝒗 , 𝒗 𝒕 = ∆𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎, 𝒙 ∈ 𝛀 , 𝒕 > 𝟎. (KS) ቐ Summary 𝒒 > 𝒎 + 𝟐 𝑵 Our result 𝑵 ≥ 𝟑, 𝒎 ≥ 𝟏, Blow-up behavior 𝒒 = 𝟐 > 𝒎 + 𝟐 𝑵 ⟹ Result 1 𝒖 ∶ radially symmetric Blow-up point is only the origin Result 2 Singularity like the Dirac delta function ⟹ In this talk we considered the following degenerate Keller—Segel system: , 𝒖 satisfies some condition Finite-time blow-up
  • 26.
  • 27. 定義 (weak solution) 𝛀 × 𝟎, 𝑻 で定義された球対称な非負値関数の組 𝒖, 𝒗 で次を 満たすものを(KS)の 𝟎, 𝑻 上の弱解という: 𝒖 ∈ 𝑳∞ 𝟎, 𝑻; 𝑳∞ 𝛀 , 𝒖 𝒎 ∈ 𝑳∞ 𝟎, 𝑻; 𝑯 𝟏 𝛀 , 𝒖 𝒎+𝟏 𝟐 ∈ 𝑯 𝟏 𝟎, 𝒕; 𝑳 𝟐 𝛀 , 𝒗 ∈ 𝑳∞ 𝟎, 𝑻; 𝑾 𝟏,∞ 𝛀 , 𝒗 𝒕 ∈ 𝑳 𝟐 𝟎, 𝑻; 𝑳 𝟐 𝛀 . ∀𝝋 ∈ 𝑳 𝟏 𝟎, 𝑻; 𝑯 𝟏 𝛀 ∩ 𝑾 𝟏,𝟏 𝟎, 𝑻; 𝑳 𝟐 𝛀 with supp𝝋 𝒙 ⊂ 𝟎, 𝑻 ; න 𝟎 𝑻 න 𝛀 (𝛁𝒖 𝒎 ⋅ 𝛁 𝝋 − 𝒖 𝒒−𝟏 𝛁𝒖 ⋅ 𝛁𝝋 − 𝒖𝝋 𝒕) 𝒅𝒙𝒅𝒕 = න 𝛀 𝒖 𝟎 𝒙 𝝋 𝒙, 𝟎 𝒅𝒙 , න 𝟎 𝑻 න 𝛀 (𝛁𝒗 ⋅ 𝛁 𝝋 + 𝒗𝝋 − 𝒖𝝋 − 𝒗𝝋 𝒕) 𝒅𝒙𝒅𝒕 = න 𝛀 𝒗 𝟎 𝒙 𝝋 𝒙, 𝟎 𝒅𝒙 . ∃𝑲 > 𝟎 s.t. 𝟐𝒆−𝟐𝒕 𝒎 + 𝟏 𝟐 න 𝟎 𝒕 න 𝛀 𝝏 𝝏𝒔 𝒖 𝒎+𝟏 𝟐 𝟐 𝒅𝒙𝒅𝒔 + 𝟏 𝟐𝒎 න 𝛀 𝛁𝒖 𝒎 𝒕 𝟐 𝒅𝒙 ≤ 𝑲, a.a. 𝒕 ∈ 𝟎, 𝑻 . for all 𝒕 < 𝑻. ただし 𝑲 は 𝒖 𝟎 𝑳 𝟐, 𝛁𝒖 𝟎 𝒎 𝑳 𝟐, 𝒗 𝟎 𝑾 𝟏,∞, 𝒖 𝑳∞(𝟎,𝑻;𝑳∞(𝛀)), 𝒎, 𝒒, 𝑵, 𝛀 に依存する定数. 6/15
  • 28. 𝒖 𝒕 = 𝛁 ⋅ 𝝓 𝒖 𝛁𝒖 − 𝝍 𝒖 𝛁𝒗 , 𝒗 𝒕 = 𝚫𝒗 − 𝒗 + 𝒖 , 𝒙 ∈ 𝛀, 𝒕 > 𝟎, 𝒙 ∈ 𝛀, 𝒕 > 𝟎. に対するcritical-condition [Winkler (2009)] (E) (E) ∃𝒔 𝟎 > 𝟏 ∃𝜺 ∈ 𝟎, 𝟏 ∃𝑲, 𝒌 > 𝟎 s.t. න 𝒔 𝟎 𝒔 𝝈𝝓 𝝈 𝝍 𝝈 𝒅𝝈 ≤ 𝑲 𝒔 𝐥𝐨𝐠 𝒔 , 𝑵 − 𝟐 − 𝜺 𝑵 න 𝒔 𝟎 𝒔 න 𝒔 𝟎 𝝈 𝝓 𝝉 𝝍 𝝉 𝒅𝝉𝒅𝝈 + 𝑲𝒔, න 𝒔 𝟎 𝒔 න 𝒔 𝟎 𝝈 𝝓 𝝉 𝝍 𝝉 𝒅𝝉𝒅𝝈 ≤ 𝒌𝒔 𝐥𝐨𝐠 𝒔 𝜽 𝒌𝒔 𝟐−𝜶 if 𝑵 = 𝟐, if 𝑵 ≥ 𝟑, if 𝑵 = 𝟐, if 𝑵 ≥ 𝟑, with some 𝜽 ∈ 𝟎, 𝟏 , with some 𝜶 > 𝟐 𝑵 , 補足スライド(一般のKeller—Segel 系について) for all 𝒔 ≥ 𝒔 𝟎. ቐ 非有界な(E)の解を与える 初期値 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑪∞ 𝛀 𝟐 が存在
  • 29. 本研究で得られたこと 有限時刻で爆発する弱解を与える 初期値が存在する. 𝒒 < 𝒎 + 𝟐 𝑵 𝒒 > 𝒎 + 𝟐 𝑵 𝒖 𝒕 = 𝚫𝒖 𝒎 − 𝛁 ⋅ (𝒖 𝒒−𝟏 𝛁𝒗), 𝒗 𝒕 = 𝚫𝒗 − 𝒗 + 𝒖, 𝒙 ∈ 𝛀, 𝒕 > 𝟎, 𝒙 ∈ 𝛀, 𝒕 > 𝟎. (KS) 補足スライド(先行研究の詳細1) 𝟐 = 𝒎 + 𝟐 𝑵 , 𝒖 𝟎 𝑳 𝟏 < ∃𝑴 𝒄 𝟐 = 𝒎 + 𝟐 𝑵 , 𝒖 𝟎 𝑳 𝟏 > ∃𝑴 𝒄 Blanchet—Laurençot (2013) (𝛀 = ℝ 𝑵 の場合) (𝛀 = ℝ 𝑵 の場合) Laurençot—Mizoguchi (2017) 時間大域的弱解が存在する. 有限時刻で爆発する 弱解を与える初期値が存在する. 𝒒 = 𝟐 , ቐ 𝒒 = 𝟐 , 𝑵 = 𝟑 or 𝟒, Ishida—Yokota (2012) (𝛀 = ℝ 𝑵 の場合) 𝒒 ≥ 𝒎 + 𝟐 𝑵 , 𝒖 𝟎, ∆𝒗 𝟎 : enough small 時間大域的弱解が存在する. Ishida—Yokota (2012), Ishida—Seki—Yokota (2014) 時間大域的弱解が存在し, 一様に有界.
  • 30. 補足スライド(先行研究の詳細2) Keller—Segel 系の有限時刻爆発についての先行研究 𝒖 𝒕 = 𝚫𝒖 𝒎 − 𝛁 ⋅ (𝒖 𝒒−𝟏 𝛁𝒗) 𝒖 𝒕 = 𝚫𝒖 − 𝛁 ⋅ ( 𝒖 𝛁𝒗) 𝒖 𝒕 = 𝚫 𝒖 + 𝜺 𝒎 − 𝛁 ⋅ ( 𝒖 + 𝜺 𝒒−𝟐 𝒖 𝛁𝒗) 𝟐 = 𝒎 + 𝟐 𝑵 , 𝒖 𝟎 𝑳 𝟏 > ∃𝑴 𝒄 Laurençot—Mizoguchi (2017) 有限時刻で爆発する 弱解を与える初期値が存在する. 𝒒 = 𝟐 , 𝑵 = 𝟑 or 𝟒, 本研究で得られたこと 有限時刻で爆発する弱解を与える 初期値が存在する.𝒒 > 𝒎 + 𝟐 𝑵 Winkler (2013) 𝑵 ≥ 𝟑 Cieslak—Stinner (2012, 2014) 有限時刻爆発解を与える初期値が存在する.𝒒 > 𝒎 + 𝟐 𝑵 ´ 有限時刻爆発解を与える初期値が存在する.
  • 31. 補足スライド(爆発解の挙動について) 結果1, 2 𝑵 ≥ 3, 𝒎 ≥ 𝟏, 𝒒 = 𝟐 とし, 𝒎, 𝒒 は を満たすものとする.また, (𝒖, 𝒗)を有限時刻 𝑻 で爆発する(KS)の弱解とする. 𝒒 = 𝟐 > 𝒎 + 𝟐 𝑵 (i) 𝒖 ∈ 𝑳∞ 𝟎, 𝑻; 𝑳 (𝛀) 𝑵(𝟐 − 𝒎) 𝟐 𝒖 は原点においてのみ爆発する.⟹ , 𝒖, 𝒗 は球対称 (ii) 𝐥𝐢𝐦 𝒕→𝑻 න 𝛀 𝒖 𝒙, 𝒕 𝝍 𝒙 𝒅𝒙 𝑵(𝟐 − 𝒎) 𝟐 が存在するすべての 𝝍 ∈ 𝑪 𝒄 ℝ 𝑵 に対して ⟹ 𝒖 は爆発点 𝒙 𝟎 において 次の意味でデルタ関数的な特異性をもつ: ∃𝑴 𝒙 𝟎 > 𝟎 ∃𝑹 > 𝟎 ∃ 𝒕 𝒏 ⊂ 𝟎, 𝑻 with 𝒕 𝒏 → 𝑻 as 𝒏 → ∞, ∃𝒇 ∈ 𝑳 𝟏 𝑩 𝒙 𝟎, 𝑹 ∩ 𝛀 ∩ 𝑳∞ 𝑨 𝒙 𝟎, 𝒓, 𝑹 ∩ 𝛀 for all 𝒓 ∈ (𝟎, 𝑹) s.t., ※ただし 𝜹 ⋅ は, Diracのデルタ関数とする. 𝒖 𝒕 𝒏 → 𝑴 𝒙 𝟎 𝜹 ⋅ −𝒙 𝟎 + 𝒇 weakly* in 𝑳∞ 𝑩 𝒙 𝟎, 𝑹 . 𝑵(𝟐 − 𝒎) 𝟐 𝒏 → ∞
  • 32. 補足スライド(有限時刻爆発解を与える初期値について) 𝑩 𝑴, 𝑨 ≔ 𝒖 𝟎, 𝒗 𝟎 ∈ 𝑳∞ 𝛀 × 𝑾 𝟏,∞ 𝛀 𝛁𝒖 𝟎 𝒎 ∈ 𝑳 𝟐 𝛀 , 𝑮 𝒖 𝟎 ∈ 𝑳 𝟏 𝛀 , 𝒖 𝟎, 𝒗 𝟎 ∶ 非負の球対称関数,න 𝛀 𝒖 𝟎 = 𝑴 , 𝒗 𝟎 𝑯 𝟏(𝛀) ≤ 𝑨, 𝑭 𝒖 𝟎, 𝒗 𝟎 ≤ −𝑲(𝑴, 𝑨) ቄ ቄ 𝑴 > 𝟎 とする. このとき,次で定められた 𝒖 𝜼, 𝒗 𝜼 に対し,∃𝜼 𝟎 > 𝟎 s.t. 𝒖 𝜼, 𝒗 𝜼 ∈ 𝑩 𝑴, 𝑨 ∀𝜼 ∈ 𝟎, 𝜼 𝟎 : 𝒖 𝜼 𝒙 ≔ 𝒂 𝜼 ⋅ 𝜼 𝜷−𝑵 𝒙 𝟐 + 𝜼 𝟐 − 𝜷 𝟐, 𝒗 𝜼 𝒙 ≔ 𝜼 𝜹−𝜸 𝒙 𝟐 + 𝜼 𝟐 − 𝜹 𝟐, if 𝑵 ≥ 𝟑. if 𝑵 = 𝟐,𝐥𝐨𝐠 𝑹 𝜼 −𝜿 𝐥𝐨𝐠 𝑹 𝟐 𝒙 𝟐 + 𝜼 𝟐 , ൞ ただし, 𝜷 > 𝑵, 𝜸 ∈ 𝟏 − 𝒒 + 𝒎 𝑵, 𝑵 − 𝟐 , 𝜹 > 𝑵 𝟐 , 𝒂 𝜼 ≔ 𝜼 𝑵−𝜷 𝑴 ‫׬‬𝛀 𝒙 𝟐 + 𝜼 𝟐 Τ−𝜷 𝟐 𝒅𝒙 . イメージ
  • 33. − න 𝟎 𝒓 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 = − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 − න {𝒖≤𝒔 𝟎} 𝒓 𝑵 𝒖𝒗 𝒓 Separate the interval!! න 𝑩 𝒓 𝟎 𝛁𝒗 𝟐 ≤ 𝝁 + 𝑪 𝒓 𝟎 𝒇 𝑳 𝟐 𝟐 + 𝒓 𝟎 𝒈 𝑳 𝟐 𝟐 + 𝒗 𝑳 𝟐 𝟐 + 𝟏න 𝛀 𝑮(𝒖) Key Lemma 𝒓 𝟎 ⋅ 𝑫 𝒖, 𝒗 ∃𝝁 ∈ [𝟎, 𝟐) ∃𝑪 > 𝟎 s.t. Key Point!! We can apply the known method − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖𝒗 𝒓 = − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖 𝒎−𝒒+𝟏 𝒖 𝒓 − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖 𝟑−𝒒 𝟐 𝒈 =: 𝑰 𝟏 + 𝑰 𝟐 𝑰 𝟏 = − න 𝒖≥𝒔 𝟎 𝒓 𝑵 𝒖 𝒎−𝒒+𝟏 𝒖 𝒓 = 𝑵 න 𝒖≥𝒔 𝟎 𝒓 𝑵−𝟏 𝑯 𝒖 − න 𝝏 𝒖≥𝒔 𝟎 𝒓 𝑵 𝑯(𝒖) 𝒅𝑺 Integration by parts 𝑯 𝒖 ≔ න 𝒔 𝟎 𝒖 𝝈 𝒎−𝒒+𝟏 𝒅𝝈where 𝒔 𝟎 𝐎 𝒓 𝟎 𝒓 ① ② ③ Collecting values ①, ② and ③, we see that 𝑰 𝟏 ≤ 𝑵 න 𝒖≥𝒔 𝟎 𝒓 𝑵−𝟏 𝑯 𝒖