Tohoku Mathematical Journal
2019
June
SECOND SERIES VOL. 71, NO. 2

Tohoku Math. J.
71 (2019), 243-279

Title STEADY STATES OF FITZHUGH-NAGUMO SYSTEM WITH NON-DIFFUSIVE ACTIVATOR AND DIFFUSIVE INHIBITOR

Author Ying Li, Anna Marciniak-Czochra, Izumi Takagi and Boying Wu

(Received January 4, 2017, revised May 11, 2017)
Abstract. In this paper, we consider a diffusion equation coupled to an ordinary differential equation with FitzHugh-Nagumo type nonlinearity. We construct continuous spatially heterogeneous steady states near, as well as far from, constant steady states and show that they are all unstable. In addition, we construct various types of steady states with jump discontinuities and prove that they are stable in a weak sense defined by Weinberger.The results are quite different from those for classical reaction-diffusion systems where all species diffuse.

Mathematics Subject Classification. Primary 35B36; Secondary 35K57, 35B35.
Key words and phrases. FitzHugh-Nagumo model, reaction-diffusion-ODE system, pattern formation, bifurcation analysis, steady states, global behaviour of solution branches, instability.

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