-
Previous Article
Wiener-Landis criterion for Kolmogorov-type operators
- DCDS Home
- This Issue
-
Next Article
Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources
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 |
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 [
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. 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. |
[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. |
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.
|
[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. 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. |
[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. |
[1] |
Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172 |
[2] |
Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134 |
[3] |
Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001 |
[4] |
Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405 |
[5] |
Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170 |
[6] |
Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053 |
[7] |
Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020319 |
[8] |
Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283 |
[9] |
Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329 |
[10] |
El Haj Laamri, Michel Pierre. Stationary reaction-diffusion systems in $ L^1 $ revisited. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 455-464. doi: 10.3934/dcdss.2020355 |
[11] |
Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334 |
[12] |
Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164 |
[13] |
Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266 |
[14] |
Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021018 |
[15] |
Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020354 |
[16] |
Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160 |
[17] |
Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 |
[18] |
Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020316 |
[19] |
Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020321 |
[20] |
Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]