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姓名 陳俞碩(Yu-Shuo Chen)  查詢紙本館藏   畢業系所 數學系
論文名稱 具有非線性化學梯度和微小擴散的廣義生物趨向性模型的脈衝解的存在性與其不穩定性
(Existence and Instability of Traveling Pulses of Generalized Keller-Segel Equations with Nonlinear Chemical Gradients and Small Diffusions)
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摘要(中) 此篇論文,我們考慮帶有非線性交互作用以及微小的擴散運動 此篇論文,我們考慮帶有非線性交互作用以及微小的擴散運動 此篇論文,我們考慮帶有非線性交互作用以及微小的擴散運動 此篇論文,我們考慮帶有非線性交互作用以及微小的擴散運動 此篇論文,我們考慮帶有非線性交互作用以及微小的擴散運動 此篇論文,我們考慮帶有非線性交互作用以及微小的擴散運動 此篇論文,我們考慮帶有非線性交互作用以及微小的擴散運動 此篇論文,我們考慮帶有非線性交互作用以及微小的擴散運動 此篇論文,我們考慮帶有非線性交互作用以及微小的擴散運動 此篇論文,我們考慮帶有非線性交互作用以及微小的擴散運動 的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統的 廣義微生物趨向性模型。我們利用奇異擾動法來證明此系統行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 行波解的存在性,我們也用到了動態系統中龐加萊理論。不考 慮微生物有擴散運動的時候 慮微生物有擴散運動的時候 慮微生物有擴散運動的時候 慮微生物有擴散運動的時候 ,我們透過分析其對應的代數微系統 ,我們透過分析其對應的代數微系統 ,我們透過分析其對應的代數微系統 ,我們透過分析其對應的代數微系統 ,我們透過分析其對應的代數微系統 ,我們透過分析其對應的代數微系統 ,我們透過分析其對應的代數微系統 給出 了脈衝解的存在性必要條件。 脈衝解的存在性必要條件。 脈衝解的存在性必要條件。 脈衝解的存在性必要條件。 脈衝解的存在性必要條件。 脈衝解的存在性必要條件。 在奇異擾動法的理論保證之 奇異擾動法的理論保證之 奇異擾動法的理論保證之 奇異擾動法的理論保證之 下, 此必要條件一樣適用於當微生物有小的擴散運動形況。同 此必要條件一樣適用於當微生物有小的擴散運動形況。同 此必要條件一樣適用於當微生物有小的擴散運動形況。同 此必要條件一樣適用於當微生物有小的擴散運動形況。同 此必要條件一樣適用於當微生物有小的擴散運動形況。同 此必要條件一樣適用於當微生物有小的擴散運動形況。同 此必要條件一樣適用於當微生物有小的擴散運動形況。同 此必要條件一樣適用於當微生物有小的擴散運動形況。同 此必要條件一樣適用於當微生物有小的擴散運動形況。同 此必要條件一樣適用於當微生物有小的擴散運動形況。同 此必要條件一樣適用於當微生物有小的擴散運動形況。同 時我們藉由譜分析 時我們藉由譜分析 時我們藉由譜分析 來討論利用上述奇異擾動法找出的脈衝解線 來討論利用上述奇異擾動法找出的脈衝解線 來討論利用上述奇異擾動法找出的脈衝解線 來討論利用上述奇異擾動法找出的脈衝解線 來討論利用上述奇異擾動法找出的脈衝解線 來討論利用上述奇異擾動法找出的脈衝解線 來討論利用上述奇異擾動法找出的脈衝解線 來討論利用上述奇異擾動法找出的脈衝解線 性穩定。 性穩定。
摘要(英) In this paper, we consider a generalized model of 2  2 Keller-Segel system with
nonlinear chemical gradient and small cell di usion. The existence of the traveling pulses
for such equations is established by the methods of geometric singular perturbation (GSP
in short) and trapping regions from dynamical systems theory. By the technique of GSP,
we show that the necessary condition for the existence of traveling pulses is that their
limiting pro les with vanishing di usion can only consist of the slow
ows on the critical
manifold of the corresponding algebraic-di erential system. We also consider the linear
stability of these pulses by the spectral analysis of the linearized operators.
關鍵字(中) ★ 趨向性
★ 奇異擾動法
★ 特徵值
★ 行波解
★ 譜分析
★ 本質譜
關鍵字(英)
論文目次 1 Introduction 1
1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Keller-Segel System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Survey on Keller-Segel System . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 Signal-dependent sensitivity . . . . . . . . . . . . . . . . . . . . . 6
1.4.2 Density-dependent sensitivity . . . . . . . . . . . . . . . . . . . . 7
1.4.3 Nonlinear di usion . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.4 Saturating signal production . . . . . . . . . . . . . . . . . . . . . 8
1.4.5 Nonlinear gradient models . . . . . . . . . . . . . . . . . . . . . . 9
1.4.6 The cell kinetics model . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Traveling Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Geometric Singular Perturbation . . . . . . . . . . . . . . . . . . . . . . 12
1.6.1 Invariant manifold theorems . . . . . . . . . . . . . . . . . . . . . 13
1.6.2 Geometric singular perturbation theory beyond normal hyperbolicity 17
2 Traveling Waves Solutions to Generalized Keller-Segel equations 20
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Traveling Wave solutions to Keller-Segel equations . . . . . . . . . . . . . 21
2.2.1 A dynamical system formulation . . . . . . . . . . . . . . . . . . . 21
2.2.2 Critical manifold, and limiting fast dynamics . . . . . . . . . . . . 23
2.2.3 The limiting slow dynamics . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 The trapping region . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 s   p0q and 1 ¡ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.1 The limiting slow dynamics . . . . . . . . . . . . . . . . . . . . . 33
2.3.2 The trapping region . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.1 Weighted function spaces . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Generalized Riemann problem of degenerate Keller-Segel systems (work-
ing in progress). 46
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A Appendix 60
A.1 Resolvent estimates for the operator L . . . . . . . . . . . . . . . . . . . 60
References 64
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指導教授 洪盟凱 審核日期 2017-7-18
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