Global dynamics on one-dimensional excitable media. (English) Zbl 1486.35114
Authors’ abstract: The FitzHugh-Nagumo system has been studied extensively for several decades. It has been shown numerically that pulses are generated to propagate and then some of the pulses are annihilated after collision. For the mathematical understanding of these complicated dynamics, we investigate the global dynamics of a one-dimensional free boundary problem in the singular limit of a FitzHugh-Nagumo type reaction-diffusion system. By introducing the notion of symbolic dynamics, we show that the asymptotic behaviors of solutions are classified into three categories: (i) the solution converges uniformly to the resting state; (ii) the solution converges to a series of traveling pulses propagating in either the same direction or both directions; and (iii) the solution converges to a propagating wave consisting of multiple traveling pulses and two traveling fronts propagating in both directions.
Reviewer: Thomas Giletti (Vandœuvre-lès-Nancy)
MSC:
| 35C07 | Traveling wave solutions |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B25 | Singular perturbations in context of PDEs |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35K57 | Reaction-diffusion equations |
| 35R35 | Free boundary problems for PDEs |
| 37B10 | Symbolic dynamics |
References:
| [1] | S. Alonso, F. Sagues, and A. S. Mikhailov, Taming Winfree turbulence of scroll waves in excitable media, Science, 299 (2003), pp. 1722-1725. |
| [2] | E. Ben-Jacob and P. Garik, The formation of patterns in non-equilibrium growth, Nature, 343 (1990), pp. 523-530. |
| [3] | E. Ben-Jacob, O. Schochet, A. Tenenbaum, I. Cohen, A. Czirok, and T. Vicsek, Generic modelling of cooperative growth patterns in bacterial colonies, Nature, 368 (1994), pp. 46-49. |
| [4] | X. Y. Chen, Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J., 21 (1991), pp. 47-83. · Zbl 0784.35048 |
| [5] | X. F. Chen, Generation and propagation of interfaces for reaction-diffusion systems, Trans. Amer. Math. Soc., 334 (1992), pp. 877-913. · Zbl 0785.35006 |
| [6] | X. F. Chen and C. Gao, Well-posedness of a free boundary problem in the limit of slow-diffusion fast-reaction systems, J. Partial Differ. Equ., 19 (2006), pp. 48-79. · Zbl 1095.35053 |
| [7] | Y.-Y. Chen, Y. Kohsaka, and H. Ninomiya, Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), pp. 697-714. · Zbl 1292.35022 |
| [8] | Y.-Y. Chen, H. Ninomiya, and R. Taguchi, Traveling spots on multi-dimensional excitable media, J. Elliptic Parabol. Equ., 1 (2015), pp. 281-305. · Zbl 1383.35103 |
| [9] | Y.-Y. Chen, H. Ninomiya, and C.-H. Wu, Global Existence and Uniqueness of Solutions for One-dimensional Reaction-interface Systems, submitted, https://arxiv.org/abs/2104.04971. |
| [10] | A. Ducrot, T. Giletti, and H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations, Trans. Amer. Math. Soc., 366 (2014), pp. 5541-5566. · Zbl 1302.35209 |
| [11] | P. C. Fife, Propagator-controller systems and chemical patterns, in Non-equilibrium Dynamics in Chemical Systems, C. Vidal and A. Pacault, eds., Springer, Berlin, 1984, pp. 76-88. · Zbl 0567.58001 |
| [12] | Y. Giga, S. Goto, and H. Ishii, Global existence of weak solutions for interface equations coupled with diffusion equations, SIAM J. Math. Anal., 23 (1992), pp. 821-835. · Zbl 0754.35061 |
| [13] | A. Hagberg and E. Meron, Pattern formation in non-gradient reaction-diffusion systems: The effects of front bifurcations, Nonlinearity, 7 (1994), pp. 805-835. · Zbl 0804.35058 |
| [14] | D. Hilhorst, Y. Nishiura, and M. Mimura, A free boundary problem arising in some reacting-diffusing system, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), pp. 355-378. · Zbl 0752.35093 |
| [15] | R. Kapral and K. Showalter, eds., Chemical Waves and Patterns, Underst. Chem. Reactiv. 10, Springer, Dordrecht, The Netherlands, 1995. |
| [16] | K. Krischer and A. Mikhailov, Bifurcation to traveling spots in reaction-diffusion systems, Phys. Rev. Lett., 73 (1994), pp. 3165-3168. |
| [17] | E. Meron, Pattern formation in excitable media, Phys. Rep., 218 (1992), pp. 1-66. |
| [18] | H. Ninomiya and C.-H. Wu, Traveling curved waves in two dimensional excitable media, SIAM J. Math. Anal., 49 (2017), pp. 777-817. · Zbl 1361.35088 |
| [19] | L. M. Pismen, Nonlocal boundary dynamics of traveling spots in a reaction-diffusion system, Phys. Rev. Lett., 86 (2001), pp. 548-551. |
| [20] | J. Rauch and J. Smoller, Qualitative theory of the FitzHugh-Nagumo equations, Adv. Math., 27 (1978), pp. 12-44. · Zbl 0379.92002 |
| [21] | J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation, Biophys. J., 13 (1973), pp. 1313-1337. |
| [22] | J. J. Tyson and J. P. Keener, Singular perturbation theory of traveling waves in excitable media (a review), Phys. D, 32 (1988), pp. 327-361. · Zbl 0656.76018 |
| [23] | A. T. Winfree, Spiral waves of chemical activity, Science, 175 (1972), pp. 634-636. |
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