非線性橢圓型偏微分方程系統之解結構分析

近年來，偏微分方程系統(以下簡稱為PDE系統)的應用非常廣泛，舉凡任何自然科學之研究，諸如物理、化學、生物等等，均與PDE系統有著相當密切的關聯性。除了蘊含較為豐富的結構性之外，PDE系統之於單一方程，亦更能刻劃與契合現實環境的條件和結果。而對於橢圓型PDE系統之探討，最普遍且主要的應用乃是有關拋物型PDE系統或反應擴散系統，其穩定態之相關研究範疇。本論文主要分為四大部分(完整內容詳見英文版論文正文)，均為探討不同類型之橢圓型PDE系統，其解結構之相關定性分析與研究。第一部分為關於所謂「次線性」合作系統之正解特性，當其非線性項於原點為一次連續可微時，我們應用「廣義隱函數定理」和「分歧理論」，證得其正解對於參數之存在性與唯一性；當非線性項於原點僅是赫爾德(Hölder)連續時，上述方法已無法直接運用，而我們得利於「上下解法」、「格林(Green)函數」，以及「極大值原理」等PDE估計方法和技巧，亦可獲致上述結論。第二部分所考慮之PDE系統為「類薛丁格(Schrödinger)」型態，除了可視為單一駐波型薛丁格方程之自然(數學型態上)推廣外，亦為所謂「漢米爾頓系統」(Hamiltonian system)之一。利用探討其相對應之「線性化」系統，以及觀察其非線性項之特性，我們能夠依據初始值進行不同解型態之分類，並對解結構全盤了解。第三部分所探討的為賦奇異性之「類劉維爾(Liouville)」型態系統，其中除了討論狄里克雷(Dirichlet)邊值問題之解的存在性與唯一性外，對於解與其依賴於非線性項之所謂「類流通量」的關係，亦有詳盡之刻劃和分類。最後，描述耗散型靜態電漿模型所衍生之「類班奈特(Bennett)」系統，為第四部分所關注之主題。對於此賦奇異性之系統，我們考慮其狄里克雷邊值問題，並對其它型態的解作完整分類。此外，我們亦證得此系統具「有限時刻爆破(finite-time blow-up)」之現象產生。

關鍵字

解結構；橢圓型偏微分方程系統

並列摘要

In the last two decades, significant progress has been made in the theory of nonlinear systems of partial differential equations. There has been much effort dedicated to the developments, motivated by applications to the natural sciences such as physics, chemistry and biology. There is also a substantial amount of consequences related to topical issues for the study of medical and ecological sciences, and other applications involving fluid, plasma, reactor dynamics and so on. In particular, systems of elliptic partial differential equations are mainly encountered in stationary problems of the theory of heat and mass transfer in reacting media, the theory of chemical reactors, combustion theory, mathematical biology and biophysics etc. This dissertation, consisting of four main parts, is devoted to studying some qualitative properties of solutions, such as the existence, uniqueness and structure of solutions to four specific kinds of nonlinear elliptic systems. In Part 1, sublinear elliptic systems are considered, and the existence, uniqueness and stability of solutions are derived via the bifurcation theory and monotonicity method. Part 2 deals with a cooperative Hamiltonian system, which can also be viewed as a natural extension of the single equation arising from investigating the stationary states of the nonlinear Schrödinger equation. By applying linearization techniques and the implicit function theorem, a complete structure of solutions is clarified. Part 3 studies a Liouville-type system with singularity at the origin. The existence and uniqueness of solutions to the Dirichlet boundary value problem are established. In addition, the structure of solutions in terms of some specific quantities, which are viewed as the total curvature or flux in the single case, will be provided as well. Finally, a singular Bennett-type system coming from modeling dissipative stationary plasmas is introduced in Part 4, and its blow-up phenomena will be investigated.

並列關鍵字

Structure of Solutions ； Elliptic PDE System

參考文獻

[1] J.-L. Chern, Z.-Y. Chen and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Comm. Math. Phys. 296 (2010), 323-351.

[2] Z.-Y. Chen, J.-L. Chern and Y.-L. Tang, On the solutions to a Liouville-type system involving singularity, Calc. Var. PDE (2011), to appear.

[3] Z.-Y. Chen, J.-L. Chern, J. Shi and Y.-L. Tang, On the uniqueness and structure of solutions to a coupled elliptic system, J. Differential Equations 249 (2010), 3419-3442.

[4] J.-L. Chern, Z.-Y. Chen, Y.-L. Tang and C.-S. Lin, Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system, J. Differential Equations 246 (2009), 3704-3714.

[5] S. Chanillo and M. K.-H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal. 5 (1995), no. 6, 924-947.

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非線性橢圓型偏微分方程系統之解結構分析

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