指導教授：陳俊全臺灣大學：數學研究所呂居謙Lu, Chu-ChienChu-ChienLu2014-11-302018-06-282014-11-302018-06-282014http://ntur.lib.ntu.edu.tw//handle/246246/264026本文探討由建造上下解以及使用單調疊代法來建構的高維度的Allen-Cahn方程式 ut = DΔ u + f(u)的非平面行波解，並透過此種方法來討論其自由邊界問題的上下解以及金字塔型的的存在之可能性.In this thesis, we discuss the non-planar travelling wave solution for the Allen-Cahn reaction-diffusion equation in high dimension ut = D Δu + f(u) by constructing super-sub solutions and use the monotonic iteration method. Also, we use this method to discuss the super-solution and sub-solution for free boundary problem and also the possibility of the existence of a pyramidal-shaped solution.致謝 i 中文摘要 ii 英文摘要 iii 1 Introduction 1 1.1 Reaction-Di usion equation: the travelling wave solution for bistable case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The free boundary problem, Stefan Condition . . . . . . . . . . . . . 2 2 Pyramidal Shaped Travelling Fronts 3 2.1 Monotone Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Pyramid and the weak sub-solution . . . . . . . . . . . . . . . . . . . 6 2.3 Super-solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Discussion on Free Boundary Problem 15 3.1 Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Spreading-Vanishing Dichotomy . . . . . . . . . . . . . . . . . . . . . 16 3.3 Survey on Pyramidal Shaped Solution . . . . . . . . . . . . . . . . . 17 參考書目 20387262 bytesapplication/pdf論文公開時間：2017/07/29論文使用權限：同意有償授權(權利金給回饋學校)反應擴散方程行波觧金字塔型自由邊界問題thesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/264026/1/ntu-103-R01221003-1.pdf