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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

Takashi Kajiwara

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.

Keywords: Sub-supersolution method, FitzHugh-Nagumo type reaction diffusion systems, heteroclinic solution.
Mathematics Subject Classification: Primary: 34C37, 35K57; Secondary: 35J50.
Citation: Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems, 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.   Google Scholar

[2]

T. Cazenave, Semilinear Schorödinger Equations, Courant Lecture Notes; 10, American Mathematical Society, 2003.  Google Scholar

[3]

C. N. Chen, S. Ei and Y. Morita, Weakly interacting wavefront dynamics in FitzHugh-Nagumo systems, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations, Vol. 19 of Grad. Stud. Math., American Mathematical Society, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. Nakashima, Stable transition layers in a balanced bistable equation, Differential and Integral Equations, 13 (2000), 1025-1038.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.   Google Scholar

[2]

T. Cazenave, Semilinear Schorödinger Equations, Courant Lecture Notes; 10, American Mathematical Society, 2003.  Google Scholar

[3]

C. N. Chen, S. Ei and Y. Morita, Weakly interacting wavefront dynamics in FitzHugh-Nagumo systems, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations, Vol. 19 of Grad. Stud. Math., American Mathematical Society, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. Nakashima, Stable transition layers in a balanced bistable equation, Differential and Integral Equations, 13 (2000), 1025-1038.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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