考慮格狀系統中FitzHugh-Nagumo類型的反應擴散方程。在這個系統，我們研究了不同擴散中的活化因子在沒有時間影響下的解。利用變分法(variational method)，我們得到一個異質分佈式在沒有時間影響下的解。而這種非常數解 (non-constant solution) 可以被看作是從trivial solution上的分歧。

A lattice dynamical system of FitzHugh-Nagumo type is considered. In this system, we study the stationary solutions under different diffusivity for activator. Using the variational method, we obtain a heterogeneous distributed stationary solution. Such a non-constant solution can be viewed as a bifurcation from the trivial solution.

1 Introduction 3
2 Related properties for matrix A 6
3 Trignometry identities 10
4 Proof of main results 13
5 References 20

[1] M. Bode, A.W. Liehr, C.P. Schenk and H.-G. Purwins, Interaction of dissipative solitons: particle-like behaviour of localized structures in a three-component
reaction-diffusion system, Physica D 161 (2002), 45-66.
[2] C.-N. Chen and X. Hu, Stability criteria for reaction-diffusion systems with skew-gradient structure, Comm. P. D. E. 33 (2008), 189-208.
[3] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys, J. 1 (1961), 445-466.
[4] G.Klaassen and E. Mitidieri, Standing wave solutions for system derived from the FitzHugh-Nagumo equations for nerve conduction, SIAM. J. Math. Anal. 17 (1986), 74-83.
[5] S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine angelfish pomacanthus, Nature 376-31 (1995), 765-768.
[6] J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. I. R. E. 50 (1962), 2061-2070.
[7] Y. Nishiura, Far-from-Equilibrium Dynamics, Translations of Mathematical Monographes (Iwanami Series in Modern Mathematics), Volumn 209, American Math. Soc., 2002.
[8] A.W. Panfilov and A.T. Winfree, Dynamical simulations of twisted scroll rings in three-dimensional excitable media, Physica D 17 (1985), 323-330.
[9] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equation, C.B.M.S. Reg. Conf. Series in Math. No. 65, Amer.
Math. Soc., Providence, RI, 1986.
[10] C. Reinecke and G. Sweers, A positive solution on Rn to a equations of FitzHugh-Nagumo type, J. Differential Equation 153 (1999), 292-312.
[11] X. Ren and J.Wei, Nucleation in the FitzHugh-Nagumo system: Interface-spike solutions, J. Differential Equations 209 (2005), 266-301.
[12] J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, Berlin/New York, 1994.
[13] A.M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B 237 (1952), 37-72.