| Summary: | 博士 === 國立臺灣大學 === 數學研究所 === 100 === 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.
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