# American Institute of Mathematical Sciences

May  2018, 38(5): 2441-2465. doi: 10.3934/dcds.2018101

## The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity

 Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan

Received  August 2017 Published  March 2018

We construct a heteroclinic solution to the FitzHugh-Nagumo type reaction-diffusion system (FHN RD system) with heterogeneity by the sub-supersolution method due to [5]. $σ(d,γ)$ is introduced as the Rayleigh quotient corresponding to a linearized eigenvalue problem of the subsolution, where $d$ and $γ$ are parameters. The key to construct the solution is the uniform estimate for $σ(·,·)$ from below. In addition, it enables us to analyze an asymptotic behavior of the solution.

Citation: Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
##### References:
 [1] D. Bonheure and L. Sanchez, Heteroclinic orbits for some classes of second and fourth order differential equations, Handbook of Differential Equations, 3 (2006), 103-202. [2] T. Cazenave, Semilinear Schorödinger Equations, Courant Lecture Notes; 10, American Mathematical Society, 2003. [3] C. N. Chen, S. Ei and Y. Morita, Weakly interacting wavefront dynamics in FitzHugh-Nagumo systems, preprint. [4] C. N. Chen, P. van Heijster, Y. Nishiura and T. Teramoto, Localized patterns in a three-component FizHugh-Nagumo model revisited via an action functional, J. Dyn. Diff. Equat., (2016). doi: 10.1007/s10884-016-9557-z. [5] C. N. Chen, S. Y. Kung and Y. Morita, Planar standing wavefronts in the FitzHugh-Nagumo equations, SIAM J. Math. Anal., 46 (2014), 657-690. doi: 10.1137/130907793. [6] E. N. Dancer and S. Yan, A minimization problem associated with elliptic systems of FitzHugh-Nagumo type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 237-253. doi: 10.1016/j.anihpc.2003.02.001. [7] L. C. Evans, Partial Differential Equations, Vol. 19 of Grad. Stud. Math., American Mathematical Society, 2010. [8] T. Kajiwara, A heteroclinic solution to a variational problem corresponding to FitzHugh-Nagumo type reaction-diffusion system with heterogeneity, Comm. Pure Appl. Anal., 16 (2017), 2133-2156. doi: 10.3934/cpaa.2017106. [9] K. Kurata and H. Matsuzawa, Multiple stable patterns in a balanced bistable equation with heterogeneous environments, Appl. Anal., 89 (2010), 1023-1035. doi: 10.1080/00036811003717947. [10] K. Nakashima, Stable transition layers in a balanced bistable equation, Differential and Integral Equations, 13 (2000), 1025-1038. [11] Y. Nishiura, Coexistence of infinitely many stable solutions to reaction-diffusion system in the singular limit, Dynamics Reported: Expositions in Dynamical Systems, Springer, New York, 3 (1994), 25-103. doi: 10.1007/978-3-642-78234-3_2. [12] Y. Oshita, On stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh-Nagumo equations in higher dimensions, J. Differential Equations, 188 (2003), 110-134. doi: 10.1016/S0022-0396(02)00084-0.

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##### References:
 [1] D. Bonheure and L. Sanchez, Heteroclinic orbits for some classes of second and fourth order differential equations, Handbook of Differential Equations, 3 (2006), 103-202. [2] T. Cazenave, Semilinear Schorödinger Equations, Courant Lecture Notes; 10, American Mathematical Society, 2003. [3] C. N. Chen, S. Ei and Y. Morita, Weakly interacting wavefront dynamics in FitzHugh-Nagumo systems, preprint. [4] C. N. Chen, P. van Heijster, Y. Nishiura and T. Teramoto, Localized patterns in a three-component FizHugh-Nagumo model revisited via an action functional, J. Dyn. Diff. Equat., (2016). doi: 10.1007/s10884-016-9557-z. [5] C. N. Chen, S. Y. Kung and Y. Morita, Planar standing wavefronts in the FitzHugh-Nagumo equations, SIAM J. Math. Anal., 46 (2014), 657-690. doi: 10.1137/130907793. [6] E. N. Dancer and S. Yan, A minimization problem associated with elliptic systems of FitzHugh-Nagumo type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 237-253. doi: 10.1016/j.anihpc.2003.02.001. [7] L. C. Evans, Partial Differential Equations, Vol. 19 of Grad. Stud. Math., American Mathematical Society, 2010. [8] T. Kajiwara, A heteroclinic solution to a variational problem corresponding to FitzHugh-Nagumo type reaction-diffusion system with heterogeneity, Comm. Pure Appl. Anal., 16 (2017), 2133-2156. doi: 10.3934/cpaa.2017106. [9] K. Kurata and H. Matsuzawa, Multiple stable patterns in a balanced bistable equation with heterogeneous environments, Appl. Anal., 89 (2010), 1023-1035. doi: 10.1080/00036811003717947. [10] K. Nakashima, Stable transition layers in a balanced bistable equation, Differential and Integral Equations, 13 (2000), 1025-1038. [11] Y. Nishiura, Coexistence of infinitely many stable solutions to reaction-diffusion system in the singular limit, Dynamics Reported: Expositions in Dynamical Systems, Springer, New York, 3 (1994), 25-103. doi: 10.1007/978-3-642-78234-3_2. [12] Y. Oshita, On stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh-Nagumo equations in higher dimensions, J. Differential Equations, 188 (2003), 110-134. doi: 10.1016/S0022-0396(02)00084-0.
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