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The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs
Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations
Research Institute for Interdisciplinary Science, Okayama University, 3-1-1, Tsushimanaka, Kita-ku, Okayama City, 700-8530, Japan |
For a balanced bistable reaction-diffusion equation, the existence of axisymmetric traveling fronts has been studied by Chen, Guo, Ninomiya, Hamel and Roquejoffre [
References:
[1] |
H. Chan and J. Wei,
Traveling wave solutions for bistable fractional Allen–Cahn equations with a pyramidal front, J. Differential Equations, 262 (2017), 4567-4609.
doi: 10.1016/j.jde.2016.12.010. |
[2] |
X. Chen,
Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
[3] |
X. Chen,
Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.
doi: 10.1016/0022-0396(92)90146-E. |
[4] |
X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya and J.-M. Roquejoffre,
Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 369-393.
doi: 10.1016/j.anihpc.2006.03.012. |
[5] |
M. del Pino, M. Kowalczyk and J. Wei,
On De Giorgi's conjecture in dimension $N\geq 9$, Annals of Math., 174 (2011), 1485-1569.
|
[6] |
M. del Pino, M. Kowalczyk and J. Wei,
Traveling waves with multiple and nonconvex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math., 66 (2013), 481-547.
|
[7] |
P. C. Fife and J. B. McLeod,
The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal., 65 (1977), 335-361.
doi: 10.1007/BF00250432. |
[8] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. |
[9] |
C. Gui,
Symmetry of traveling wave solutions to the Allen–Cahn equation in $\mathbb{R}^{2}$, Arch. Rat. Mech. Anal., 203 (2012), 1037-1065.
doi: 10.1007/s00205-011-0480-5. |
[10] |
F. Hamel, R. Monneau and J.-M. Roquejoffre,
Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096.
doi: 10.3934/dcds.2005.13.1069. |
[11] |
F. Hamel, R. Monneau and J.-M. Roquejoffre,
Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92.
|
[12] |
M. Haragus and A. Scheel,
Corner defects in almost planar interface propagation, Ann. I. H. Poincaré, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 283-329.
doi: 10.1016/j.anihpc.2005.03.003. |
[13] |
Y. Kurokawa and M. Taniguchi,
Multi-dimensional pyramidal traveling fronts in the Allen–Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054.
doi: 10.1017/S0308210510001253. |
[14] |
W.-M. Ni and M. Taniguchi,
Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379-395.
doi: 10.3934/nhm.2013.8.379. |
[15] |
H. Ninomiya and M. Taniguchi,
Existence and global stability of traveling curved fronts in the Allen–Cahn equations, J. Differential Equations, 213 (2005), 204-233.
doi: 10.1016/j.jde.2004.06.011. |
[16] |
H. Ninomiya and M. Taniguchi,
Global stability of traveling curved fronts in the Allen–Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832.
|
[17] |
M. Taniguchi,
Traveling fronts of pyramidal shapes in the Allen–Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: 10.1137/060661788. |
[18] |
M. Taniguchi,
The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen–Cahn equations, J. Differential Equations, 246 (2009), 2103-2130.
doi: 10.1016/j.jde.2008.06.037. |
[19] |
M. Taniguchi, Pyramidal traveling fronts in the Allen–Cahn equations, RIMS Kôkyûroku, 1651 (2009), 92–109. |
[20] |
M. Taniguchi,
Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046.
doi: 10.3934/dcds.2012.32.1011. |
[21] |
M. Taniguchi,
An $(N-1)$-dimensional convex compact set gives an $N$-dimensional traveling front in the Allen–Cahn equation, SIAM J. Math. Anal., 47 (2015), 455-476.
doi: 10.1137/130945041. |
[22] |
M. Taniguchi,
Convex compact sets in $\mathbb{R}^{ N-1}$ give traveling fronts of cooperation-diffusion systems in $\mathbb{R}^{ N}$, J. Differential Equations, 260 (2016), 4301-4338.
|
[23] |
M. Taniguchi, Traveling front solutions in reaction-diffusion equations, submitted. |
[24] |
M. Taniguchi,
Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1791-1816.
doi: 10.1016/j.anihpc.2019.05.001. |
[25] |
X.-J. Wang,
Convex solutions to the mean curvature flow, Ann. of Math., 173 (2011), 1185-1239.
doi: 10.4007/annals.2011.173.3.1. |
[26] |
Z.-C. Wang,
Traveling curved fronts in monotone bistable systems, Discrete Contin. Dyn. Syst., 32 (2012), 2339-2374.
doi: 10.3934/dcds.2012.32.2339. |
show all references
References:
[1] |
H. Chan and J. Wei,
Traveling wave solutions for bistable fractional Allen–Cahn equations with a pyramidal front, J. Differential Equations, 262 (2017), 4567-4609.
doi: 10.1016/j.jde.2016.12.010. |
[2] |
X. Chen,
Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
[3] |
X. Chen,
Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.
doi: 10.1016/0022-0396(92)90146-E. |
[4] |
X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya and J.-M. Roquejoffre,
Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 369-393.
doi: 10.1016/j.anihpc.2006.03.012. |
[5] |
M. del Pino, M. Kowalczyk and J. Wei,
On De Giorgi's conjecture in dimension $N\geq 9$, Annals of Math., 174 (2011), 1485-1569.
|
[6] |
M. del Pino, M. Kowalczyk and J. Wei,
Traveling waves with multiple and nonconvex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math., 66 (2013), 481-547.
|
[7] |
P. C. Fife and J. B. McLeod,
The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal., 65 (1977), 335-361.
doi: 10.1007/BF00250432. |
[8] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. |
[9] |
C. Gui,
Symmetry of traveling wave solutions to the Allen–Cahn equation in $\mathbb{R}^{2}$, Arch. Rat. Mech. Anal., 203 (2012), 1037-1065.
doi: 10.1007/s00205-011-0480-5. |
[10] |
F. Hamel, R. Monneau and J.-M. Roquejoffre,
Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096.
doi: 10.3934/dcds.2005.13.1069. |
[11] |
F. Hamel, R. Monneau and J.-M. Roquejoffre,
Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92.
|
[12] |
M. Haragus and A. Scheel,
Corner defects in almost planar interface propagation, Ann. I. H. Poincaré, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 283-329.
doi: 10.1016/j.anihpc.2005.03.003. |
[13] |
Y. Kurokawa and M. Taniguchi,
Multi-dimensional pyramidal traveling fronts in the Allen–Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054.
doi: 10.1017/S0308210510001253. |
[14] |
W.-M. Ni and M. Taniguchi,
Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379-395.
doi: 10.3934/nhm.2013.8.379. |
[15] |
H. Ninomiya and M. Taniguchi,
Existence and global stability of traveling curved fronts in the Allen–Cahn equations, J. Differential Equations, 213 (2005), 204-233.
doi: 10.1016/j.jde.2004.06.011. |
[16] |
H. Ninomiya and M. Taniguchi,
Global stability of traveling curved fronts in the Allen–Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832.
|
[17] |
M. Taniguchi,
Traveling fronts of pyramidal shapes in the Allen–Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: 10.1137/060661788. |
[18] |
M. Taniguchi,
The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen–Cahn equations, J. Differential Equations, 246 (2009), 2103-2130.
doi: 10.1016/j.jde.2008.06.037. |
[19] |
M. Taniguchi, Pyramidal traveling fronts in the Allen–Cahn equations, RIMS Kôkyûroku, 1651 (2009), 92–109. |
[20] |
M. Taniguchi,
Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046.
doi: 10.3934/dcds.2012.32.1011. |
[21] |
M. Taniguchi,
An $(N-1)$-dimensional convex compact set gives an $N$-dimensional traveling front in the Allen–Cahn equation, SIAM J. Math. Anal., 47 (2015), 455-476.
doi: 10.1137/130945041. |
[22] |
M. Taniguchi,
Convex compact sets in $\mathbb{R}^{ N-1}$ give traveling fronts of cooperation-diffusion systems in $\mathbb{R}^{ N}$, J. Differential Equations, 260 (2016), 4301-4338.
|
[23] |
M. Taniguchi, Traveling front solutions in reaction-diffusion equations, submitted. |
[24] |
M. Taniguchi,
Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1791-1816.
doi: 10.1016/j.anihpc.2019.05.001. |
[25] |
X.-J. Wang,
Convex solutions to the mean curvature flow, Ann. of Math., 173 (2011), 1185-1239.
doi: 10.4007/annals.2011.173.3.1. |
[26] |
Z.-C. Wang,
Traveling curved fronts in monotone bistable systems, Discrete Contin. Dyn. Syst., 32 (2012), 2339-2374.
doi: 10.3934/dcds.2012.32.2339. |


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