在這篇論文中我們感興趣的是具有一個活化因子和兩個抑制因子與FitzHugh-Nagumo方程的性質有許多相似之反應擴散方程所產生行進波。藉由一個擁有非局部項的泛函以及使用截斷論證的方式得到一個行進波解的存在性。波動傳遞是由其中一個穩定平衡態逐漸趨向另一個穩定平衡態。

We are interested in the traveling wave solutions of reaction-diffusion equations. The model considered in the thesis is a FitzHugh-Nagumo type system in which one activator interacting with two inhibitors. In this system there are three equilibria, and two of them are stable. We employ variational arguments to a functional with non-local terms. With the aide of a truncation argument, we obtain a traveling front from one stable equilibrium moving towards another stable equilibrium.

[1]D.G. Aronson and H.F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Proceedings of the Tulane Program in Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, 446, pp. 549. Springer, Berlin, 1975.
[2]H. Berestycki, B. Larrouturou and P. -L. Lions, Multidimensional traveling wave solutions of a flame propagation model, Arch. Rat. Mech. Anal. 111 (1990), 33-49.
[3]H. Berestycki and L. Nirenberg, On the method of moving planes and sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), 1-37.
[4]H. Berestycki and L. Nirenberg, Traveling fronts in cylinders, Ann. Inst. H. Poincar\'e Anal. Non Linaire 9 (1992), 497-572.
[5]C. -N. Chen, C, -C Chen and C. -C Huang, Traveling waves
for the FitzHugh-Nagumo system on an infinite channel, J. Differential Equations 261 (2016), 3010-3041.
[6] C. -N. Chen and Y. S. Choi, Standing pulse solutions to FitzHugh-Nagumo equations, Arch. Rational Mech. Anal. 206 (2012), 741-777.
[7] C. -N. Chen and Y. S. Choi, Traveling pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations 54 (2015), 1-45.
[8]C. -N. Chen, Y. Choi and X. Hu, An index method for stability analysis of traveling and standing waves, preprint.
[9]C. -N. Chen, S. -I. Ei, Y. -P Lin and S. -Y Kung, Standing waves joining with Turing patterns in FitzHugh-Nagumo type systems, Communications in Partial Differential Equations 36 (2011), 998-1015.
[10] C. -N. Chen and X. Hu, Maslov index for homoclinic orbits of Hamiltonian systems, Ann. Inst. H. Poincare Anal. Non Linearie 24 (2007), 589-603.
[11] C. -N. Chen and X. Hu, Stability criteria for reaction-diffusion systems with skew-gradient structure, Communications in Partial Differential Equations 33 (2008), 189-208.
[12] C. -N. Chen and X. Hu, Stability analysis for standing pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 827-845
[13]C. -N. Chen, Shih-Yin Kung and Yoshihisa Morita, Planar standing wavefronts in the FitzHugh-Nagumo equations, SIAM J. Math. Anal. 46 (2014), 657-690.
[14]C. -N. Chen and K. Tanaka, A variational approach for standing waves of FitzHugh-Nagumo type systems, J. Differential Equations 257 (2014), 109-144.
[15] A. Doelman, P. van Heijster and T. Kaper, Pulse dynamics
in a three-component system: existence analysis, J. Dynam. Differential
Equations 21 (2008), 73-115.
[16]L. C. Evans, Partial Differential Equations, Second Edition, American Mathematical Society, 2010.
[17]P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65 (1977), 335-361.
[18] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. {\bf 1} (1961), 445-466.
[19]S. Heinze, A variational approach to traveling waves, Preprint 85, Max Planck Institute for Mathematical Sciences, 2001.
[20] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952), 500-544.
[21]C. K. R. T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc. 286 (1984), 431-469.
[22]A. W. Liehr, {\it Dissipative Solitons in Reaction-Diffusion Systems}, Springer Series in Synergetics 70, Springer-Verlag, Berlin, 2013.
[23]M. Lucia, C. Muratov and M. Novaga, Linear vs. nonlinear selection for
the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium, Communications on Pure and Applied Mathematics, 57 (2004),
616-636.
[24]J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. I. R. E. 50 (1962), 2061-2070.
[25]Y. Nishiura, T. Teramoto, X. Yuan and K.-I. Udea, Dynamics of traveling pulses in heterogeneous media, CHAOS 17(3) (2007).
[26]M. H. Protter and H.F. Weinberger, {\it Maximum Principles
in Differential Equations}, Pentice-Hall, Englewood Cliffs, 1967.
[27]J. F. Reineck, Traveling wave solutions to a gradient system. Trans. Amer. Math. Soc. 307 (1988), 535-544.
[28]C. Reinecke and G. Sweers, A positive solution on Rn to a equations of FitzHugh-Nagumo type, J. Differential Equations 153 (1999), 292-312.
[29]D. Terman, Infinitely many traveling wave solutions of a gradient system. Trans. Amer. Math. Soc. 301 (1987), 537-556 .
[30] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B 237 (1952), 37-72.
[31]J. M. Vega, The asymptotic behavior of the solutions of some semilinear elliptic equations in cylindrical domains. J. Differ. Equations 102 (1993), 119-152.
[32]A. Volpert, V. Volpert and V. Volpert, {\it Traveling Wave Solutions of Parabolic Systems. American Mathematical Society}, Providence, 1994.
[33]E. Yanagida, Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol. 22 (1985), 81-104.