在N 維空間中的費滋漢那古默系統之行波解

Abstract

本篇論文主要研究在N維空間中擴散的費滋漢那古默系統行波解之存在性。這個系統有反梯度結構亦有非局部的梯度結構。此外，藉由適當的變換它在某些參數範圍之下亦有單調系統結構。 對於有界區域，變分法用來建構此系統有邊值條件平衡態的存在性。無界柱狀區域並且當方程的擴散系數相同時，我們藉由上述的三種結構來研究行波解。使用非局部的能量來證明行波解的存在性並用能量觀點刻畫波速；另一方面，使用反梯度結構，我們給出波與波速的最大最小能量公式刻畫方法。關於全空間區域，應用上解下解法來建立在N維空間中單邊穩定形式之行波解。除此之外，藉著反射方法也建立了在1維空間中無窮多個周期駐波。

In this thesis, we study the existences of travelling waves of the diffusive FitzHugh-Nagumo system (DFHN) in R^N. This system has a skew-gradient structure as defined by Yanagida as well as a non-local gradient structure. In addition, by a suitable transformation, it also has a monotone-system structure on some parameter ranges. For bounded domains, the variational approach is applied to construct steady states of (DFHN) with Dirichlet or/and Neumann condition. For unbounded cylindrical domains, we study the travelling wave solutions via all of the three structures mentioned above when the diffusion coefficients in the equations are equal. By using the nonlocal variational energy, we establish the existence of a travelling front solution for (DFHN). Our existence result also obtains a variational characterization for the wave speed. On the other hand, using the skew-gradient structure, we give a mini-max formulation of the travelling wave and its speed. For whole domains, we employ the method of super- and subsolutions to establish the existence of monostable-type traveling wave solutions in R^N. Moreover, we construct infinitely many standing periodic solutions in R^1 based on the reflection method.