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Li, Ying; Marciniak-Czochra, Anna; Takagi, Izumi; Wu, Boying

Steady states of FitzHugh-Nagumo system with non-diffusive activator and diffusive inhibitor. (English) Zbl 1429.35130

Tohoku Math. J. (2) 71, No. 2, 243-279 (2019).
The target of the authors is a diffusion equation coupled to an ordinary differential equation with FitzHugh-Nagumo type nonlinearity. They construct continuous spatially heterogeneous steady states near, and correspondiongly far, from constant steady states. They show that these states are all unstable. They construct various types of steady states with jump discontinuities and prove that they are stable in a weak sense.
Reviewer: Adrian Muntean (Karlstad)

Cited in 4 Documents

MSC:

35K57 Reaction-diffusion equations
35B36 Pattern formations in context of PDEs
35B35 Stability in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C15 Developmental biology, pattern formation

Keywords:

FitzHugh-Nagumo model; reaction-diffusion-ODE system; pattern formation; bifurcation analysis; steady states; global behaviour of solution branches; instability
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Full Text: DOI Euclid

References:

[1] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Anal. 8 (1971), 321-340. · Zbl 0219.46015 · doi:10.1016/0022-1236(71)90015-2
[2] C. N. Chen, S. I. Ei, Y. P. Lin and S. Y. Kung, Standing waves joining with Turing patterns in FitzHugh-Nagumo type systems, Comm. Partial Differential Equations 36 (2011), 998-1015. · Zbl 1233.35114 · doi:10.1080/03605302.2010.509769
[3] 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. · Zbl 1343.35134 · doi:10.1016/j.jde.2016.05.014
[4] L. H. Chuan, T. Tsujikawa and A. Yagi, Stationary solutions to forest kinematic model, Glasgow Mathematical Journal 51(1) (2009-01), 1-17. · Zbl 1152.35366 · doi:10.1017/S0017089508004485
[5] E. N. Dancer and S. Yan, Solutions with interior and boundary peaks for the Neumann problem of an elliptic system of FitzHugh-Nagumo type, Indiana Univ. Math. J. 55 (2006), 217-258. · Zbl 1186.35053 · doi:10.1512/iumj.2006.55.2614
[6] V. Klika, R. Baker, D. Headon and E. Gaffney, The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organisation, Bull. Math. Biol. 74 (2012), 935-957. · Zbl 1235.92010 · doi:10.1007/s11538-011-9699-4
[7] S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science 329 (2010), 1616-1620. · Zbl 1226.35077 · doi:10.1126/science.1179047
[8] T. Kostova, R. Ravindran and M. Schonbek, Fitzhugh-Nagumo revisited: Types of bifurcations, periodical forcing and stability regions by a Lyapunov functional, Internati. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), 913-925. · Zbl 1100.34031 · doi:10.1142/S0218127404009685
[9] Y. Li and A. Marciniak-Czochra, I. Takagi and B. Y. Wu, Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis, Hiroshima Math. J. 47 (2017), 217-247. · Zbl 1373.35043 · doi:10.32917/hmj/1499392826
[10] A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of turing patterns in reaction-diffusion-ODE systems, J. Math. Biol. 74 (2017), 583-618. · Zbl 1356.35043 · doi:10.1007/s00285-016-1035-z
[11] A. Marciniak-Czochra, M. Nakayama and I. Takagi, Pattern formation in a diffusion-ode model with hysteresis, Differential Integral Equations 28 (2015), 655-694. · Zbl 1363.35261
[12] A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: influence of growth factor production and cooperation between partially transformed cells, Math. Models Methods Appl. Sci. 17 (2007), 1693-1719. · Zbl 1135.92019 · doi:10.1142/S0218202507002443
[13] M. Mimura, M. Tabata and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal. 11 (1980), 613-631. · Zbl 0438.34014 · doi:10.1137/0511057
[14] Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal. 13 (1982), 555-593. · Zbl 0501.35010 · doi:10.1137/0513037
[15] Y. Oshita, On stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh-Nagumo equations in higher dimensions, J. Differential Equations 188 (2003), 110-134. · Zbl 1036.35083 · doi:10.1016/S0022-0396(02)00084-0
[16] X. Ren and J. Wei, Nucleation in the FitzHugh-Nagumo system: Interface-spike solutions, J. Differential Equations 209 (2005), 266-301. · Zbl 1072.35037 · doi:10.1016/j.jde.2004.08.019
[17] C. Rocsoreanu, A. Georgescu and N. Giurgiteanu, The FitzHugh-Nagumo model: Bifurcation and dynamics, Kluwer Academic Publishers, Boston, 2000. · Zbl 0959.34031
[18] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Second edition, Springer-Verlag, New York, 1994. · Zbl 0807.35002
[19] A. M. Turing, The chemical basis of morphogenesis, Philo. Trans. Roy. Soc. London Ser. B 237 (1952), 37-72. · Zbl 1403.92034 · doi:10.1098/rstb.1952.0012
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