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November  2017, 16(6): 2133-2156. doi: 10.3934/cpaa.2017106

A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity

Takashi Kajiwara

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

Received  January 2017 Revised  July 2017 Published  July 2017

Chen, Kung and Morita [5] studied a variational problem corresponding to the FitzHugh-Nagumo type reaction-diffusion system (FHN type RD system), and they proved the existence of a heteroclinic solution to the system.

Motivated by [5], we consider a variational problem corresponding to FHN type RD system which involves heterogeneity. We prove the existence of a heteroclinic solution to the problem under certain conditions on the heterogeneity. Moreover, we give some information about the location of the transitions.

Keywords: Variational problem, FitzHugh-Nagumo type reaction diffusion systems, heteroclinic solution.
Mathematics Subject Classification: Primary: 35J50, 35K57; Secondary: 35Q92.
Citation: Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
References:
[1]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differencial Equations Springer, New York, 2010.

[2]

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.  doi: 10.1016/S1874-5725(06)80006-4.

[3]

C. N. Chen and Y. Choi, Standing pulse solutions to FitzHugh-Nagumo equations, Arch. Ration. Mech. Anal., 206 (2012), 741-777.  doi: 10.1007/s00205-012-0542-3.

[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/S0294-1449(03)00032-5.

[7]

S. Ei and H. Ikeda, Front dynamics in heterogeneous diffusive media, Physica D, 239 (2010), 1637-1649.  doi: 10.1016/j.physd.2010.04.008.

[8]

T. Kajiwara and K. Kurata, On a variational problem arising from the three-component FitzHugh-Nagumo type reaction-diffusion systems, Tokyo J. Math. accepted, 2016.

[9]

Y. Nishiura, T. Teramoto and X. Yuan, Heterogeneity-introduced spot dynamics for a three-component reaction-diffusion system, Comm. Pure Appl. Anal., 11 (2012), 307-338.  doi: 10.3934/cpaa.2012.11.307.

[10]

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.

[11]

C. Sourdis, The heteroclinic connection problem for general double-well potentials, Mediterr. J. Math., 13 (2016), 4693-4710.  doi: 10.1007/s00009-016-0770-0.

show all references

References:
[1]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differencial Equations Springer, New York, 2010.

[2]

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.  doi: 10.1016/S1874-5725(06)80006-4.

[3]

C. N. Chen and Y. Choi, Standing pulse solutions to FitzHugh-Nagumo equations, Arch. Ration. Mech. Anal., 206 (2012), 741-777.  doi: 10.1007/s00205-012-0542-3.

[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/S0294-1449(03)00032-5.

[7]

S. Ei and H. Ikeda, Front dynamics in heterogeneous diffusive media, Physica D, 239 (2010), 1637-1649.  doi: 10.1016/j.physd.2010.04.008.

[8]

T. Kajiwara and K. Kurata, On a variational problem arising from the three-component FitzHugh-Nagumo type reaction-diffusion systems, Tokyo J. Math. accepted, 2016.

[9]

Y. Nishiura, T. Teramoto and X. Yuan, Heterogeneity-introduced spot dynamics for a three-component reaction-diffusion system, Comm. Pure Appl. Anal., 11 (2012), 307-338.  doi: 10.3934/cpaa.2012.11.307.

[10]

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.

[11]

C. Sourdis, The heteroclinic connection problem for general double-well potentials, Mediterr. J. Math., 13 (2016), 4693-4710.  doi: 10.1007/s00009-016-0770-0.

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