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June  2020, 40(6): 3981-3995. doi: 10.3934/dcds.2020126

Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations

Masaharu Taniguchi

Research Institute for Interdisciplinary Science, Okayama University, 3-1-1, Tsushimanaka, Kita-ku, Okayama City, 700-8530, Japan

Received  April 2019 Revised  September 2019 Published  February 2020

For a balanced bistable reaction-diffusion equation, the existence of axisymmetric traveling fronts has been studied by Chen, Guo, Ninomiya, Hamel and Roquejoffre [4]. This paper gives another proof of the existence of axisymmetric traveling fronts. Our method is as follows. We use pyramidal traveling fronts for unbalanced reaction-diffusion equations, and take the balanced limit. Then we obtain axisymmetric traveling fronts in a balanced bistable reaction-diffusion equation. Since pyramidal traveling fronts have been studied in many equations or systems, our method might be applicable to study axisymmetric traveling fronts in these equations or systems.

Keywords: Traveling front, reaction-diffusion equation, axisymmetric, balanced.
Mathematics Subject Classification: Primary: 35C07, 35B20; Secondary: 35K57.
Citation: Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126
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.

Figure 1.  A level set $ \left\{ {U(\mathit{\boldsymbol{x'}}, {x_n}) = {\theta _0}} \right\}$ of $ U $
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Figure 2.  A level set $ \{V_{k}(\mathit{\boldsymbol{x}}', x_{n}) = \theta_{0}\} $ of a square pyramidal traveling front $ V_{k} $
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