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May 2017, 22(3): 1111-1144. doi: 10.3934/dcdsb.2017055

## On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China 2 School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, Ningxia 750021, China 3 Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA

Corresponding author

Dedicated to Professor Robert Stephen Cantrell on the occasion of his 60th birthday

Received  July 2015 Revised  April 2016 Published  December 2016

This paper is concerned with the following two-species Lotka-Volterra competition-diffusion system in the three-dimensional spatial space
 $\left\{ {\begin{array}{*{20}{l}}{\frac{\partial }{{\partial t}}{u_1}({\bf{x}},t) = \Delta {u_1}({\bf{x}},t) + {u_1}({\bf{x}},t)\left[ {1 - \;{u_1}({\bf{x}},t) - {k_1}{u_2}({\bf{x}},t)} \right],}\\{\frac{\partial }{{\partial t}}{u_2}({\bf{x}},t) = d\Delta {u_2}({\bf{x}},t) + r{u_2}({\bf{x}},t)\left[ {1 - {u_2}({\bf{x}},t) - {k_2}{u_1}({\bf{x}},t)} \right],}\end{array}} \right.$
where
 $\mathbf{x}∈ \mathbb{R}^3$
and
 $t>0$
. For the bistable case, namely
 $k_1,k_2>1$
, it is well known that the system admits a one-dimensional monotone traveling front
 $\mathbf{\Phi}(x+ct)=\left(\Phi_1(x+ct),\Phi_2(x+ct)\right)$
connecting two stable equilibria
 $\mathbf{E}_u=(1,0)$
and
 $\mathbf{E}_v=(0,1)$
, where
 $c∈\mathbb{R}$
is the unique wave speed. Recently, two-dimensional Ⅴ-shaped fronts and high-dimensional pyramidal traveling fronts have been studied under the assumption that
 $c>0$
. In this paper it is shown that for any
 $s>c>0$
, the system admits axisymmetric traveling fronts
 $\mathbf{\Psi}(\mathbf{x}^\prime,x_3+st)=\left(\Phi_1(\mathbf{x}^\prime,x_3+st),\Phi_2(\mathbf{x}^\prime, x_3+st)\right)$
in
 $\mathbb{R}^3$
connecting
 $\mathbf{E}_u=(1,0)$
and
 $\mathbf{E}_v=(0,1)$
, where
 $\mathbf{x}^\prime∈\mathbb{R}^2$
. Here an axisymmetric traveling front means a traveling front which is axially symmetric with respect to the
 $x_3$
-axis. Moreover, some important qualitative properties of the axisymmetric traveling fronts are given. When
 $s$
tends to
 $c$
, it is proven that the axisymmetric traveling fronts converge locally uniformly to planar traveling wave fronts in
 $\mathbb{R}^3$
. The existence of axisymmetric traveling fronts is obtained by constructing a sequence of pyramidal traveling fronts and taking its limit. The qualitative properties are established by using the comparison principle and appealing to the asymptotic speed of propagation for the resulting system. Finally, the nonexistence of axisymmetric traveling fronts with concave/convex level sets is discussed.
Citation: Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055
##### References:
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Chapuisat, Existence and nonexistence of curved front solution of a biological equation, J. Differential Equations, 236 (2007), 237-279. doi: 10.1016/j.jde.2007.01.021. [7] 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. Linéaire, 24 (2007), 369-393. doi: 10.1016/j.anihpc.2006.03.012. [8] C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343. doi: 10.1512/iumj.1984.33.33018. [9] D. Daners and P. K. McLeod, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes Math. Ser. 279, Longman Scientific and Technical, Harlow, 1992. [10] M. El Smaily, F. Hamel and R. Huang, Two-dimensional curved fronts in a periodic shear flow, Nonlinear Analysis TMA, 74 (2011), 6469-6486. doi: 10.1016/j.na.2011.06.030. [11] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference, Series in Applied Mathematics, 53, 1988. [12] S. A. Gardner, Existence and stability of travelling wave solutions of competition model: A degree theoretical approach, J. Differential Equations, 44 (1982), 343-364. doi: 10.1016/0022-0396(82)90001-8. [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. [14] C. Gui, Symmetry of traveling wave solutions to the Allen-Cahn equation in ℝ2, Arch. Rational Mech. Anal., 203 (2012), 1037-1065. doi: 10.1007/s00205-011-0480-5. [15] J.-S. Guo and Y.-C. Lin, The sign of the wave speed for the Lotka-Volterra competition-diffusion system, Comm. Pure Appl. Anal., 12 (2013), 2083-2090. doi: 10.3934/cpaa.2013.12.2083. [16] J.-S. Guo and C.-H. Wu, Recent developments on wave propagation in 2-species competition systems, Discrete Contin. Dynam. Syst.-B, 17 (2012), 2713-2724. doi: 10.3934/dcdsb.2012.17.2713. [17] J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533. doi: 10.1016/j.jde.2010.12.004. [18] F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in ℝN with conicalshaped level sets, Comm. Partial Differential Equations, 25 (2000), 769-819. doi: 10.1080/03605300008821532. [19] F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ecole Norm. Sup., 37 (2004), 469-506. doi: 10.1016/j.ansens.2004.03.001. [20] F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dynam. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069. [21] F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dynam. Syst., 14 (2006), 75-92. [22] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in ℝN, Arch. Rational Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238. [23] F. Hamel and J.-M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Discrete Contin. Dynam. Syst.-S, 4 (2011), 101-123. doi: 10.3934/dcdss.2011.4.101. [24] M. Haragus and A. Scheel, A bifurcation approach to non-planar traveling waves in reactiondiffusion systems, GAMM-Mitt., 30 (2007), 75-95. doi: 10.1002/gamm.200790012. [25] M. Haragus and A. Scheel, Almost planar waves in anisotropic media, Comm. Partial Differential Equations, 31 (2006), 791-815. doi: 10.1080/03605300500361420. [26] M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré Anal. Linéaire, 23 (2006), 283-329. doi: 10.1016/j.anihpc.2005.03.003. [27] R. Huang, Stability of travelling fronts of the Fisher-KPP equation in ℝN, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 599-622. doi: 10.1007/s00030-008-7041-0. [28] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556. [29] Y. Kan-on, Existence of standing waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 117-133. doi: 10.1007/BF03167302. [30] Y. Kan-on, Instability of stationary solutions for a Lotka-Volterra competition model with diffusion, J. Math. Anal. Appl., 208 (1997), 158-170. doi: 10.1006/jmaa.1997.5309. [31] Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343-349. doi: 10.1007/BF03167252. [32] Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal travelling fronts in the AllenCahn equations, Proc. Royal Soc. Edinburgh Sect. A: Math., 14 (2011), 1031-1054. doi: 10.1017/S0308210510001253. [33] W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reactiondiffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003. [34] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. [35] G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513. doi: 10.1016/j.jde.2007.10.019. [36] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. [37] Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sinica, 3 (2008), 567-584. [38] Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240. doi: 10.1137/080723715. [39] 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. [40] 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. [41] H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dynam. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819. [42] M. del Pino, M. Kowalczyk and J. Wei, A counterexample to a conjecture by De Giorgi in large dimensions, C. R. Math. Acad. Sci. Paris, 346 (2008), 1261-1266. doi: 10.1016/j.crma.2008.10.010. [43] 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. doi: 10.1002/cpa.21438. [44] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, PrenticeHall, Inc. , Englewood Cliffs, N. J. , 1967. 1144 [45] W.-J. Sheng, W.-T. Li and Z.-C. Wang, Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity, J. Differential Equations, 252 (2012), 2388-2424. doi: 10.1016/j.jde.2011.09.01. [46] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, Amer. Math. Soc. , Providence, RI, 1995. [47] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788. [48] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Euqations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037. [49] M. Taniguchi, Multi-Dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dynam. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011. [50] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [51] A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc. , Providence, RI, 1994. [52] Z.-C. Wang, Traveling curved fronts in monotone bistable systems, Discrete Contin. Dynam. Syst., 32 (2012), 2339-2374. doi: 10.3934/dcds.2012.32.2339. [53] Z.-C. Wang, Cylindrically symmetric traveling fronts in periodic reaction-diffusion equation with bistable nonlinearity, Proc. Royal Soc. Edinburgh Sect. A: Math., 145 (2015), 1053-1090. doi: 10.1017/S0308210515000268. [54] Z.-C. Wang and Z.-H. Bu, Nonplanar traveling fronts in reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearities, J. Differential Equations, 260 (2016), 6405-6450. doi: 10.1016/j.jde.2015.12.045. [55] Z.-C. Wang, W.-T. Li and S. Ruan, Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems, Sci. China Math., 59 (2016), 1868-1908. doi: 10.1007/s11425-016-0015-x. [56] Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. 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show all references

##### References:
 [1] E. O. Alcahrani, F. A. Davidson and N. Dodds, Travelling waves in near-degenerate bistable competition models, Math. Model. Nat. Phenom., 5 (2010), 13-35. doi: 10.1051/mmnp/20105502. [2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math.,, 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. [3] A. Bonnet and F. Hamel, Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80-118. doi: 10.1137/S0036141097316391. [4] P. K. Brazhnik and J. J. Tyson, On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (1999), 371-391. doi: 10.1137/S0036139997325497. [5] Z.-H. Bu and Z.-C. Wang, Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media, Commun. Pure Appl. Anal., 15 (2016), 139-160. doi: 10.3934/cpaa.2016.15.139. [6] G. Chapuisat, Existence and nonexistence of curved front solution of a biological equation, J. Differential Equations, 236 (2007), 237-279. doi: 10.1016/j.jde.2007.01.021. [7] 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. Linéaire, 24 (2007), 369-393. doi: 10.1016/j.anihpc.2006.03.012. [8] C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343. doi: 10.1512/iumj.1984.33.33018. [9] D. Daners and P. K. McLeod, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes Math. Ser. 279, Longman Scientific and Technical, Harlow, 1992. [10] M. El Smaily, F. Hamel and R. Huang, Two-dimensional curved fronts in a periodic shear flow, Nonlinear Analysis TMA, 74 (2011), 6469-6486. doi: 10.1016/j.na.2011.06.030. [11] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference, Series in Applied Mathematics, 53, 1988. [12] S. A. Gardner, Existence and stability of travelling wave solutions of competition model: A degree theoretical approach, J. Differential Equations, 44 (1982), 343-364. doi: 10.1016/0022-0396(82)90001-8. [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. [14] C. Gui, Symmetry of traveling wave solutions to the Allen-Cahn equation in ℝ2, Arch. Rational Mech. Anal., 203 (2012), 1037-1065. doi: 10.1007/s00205-011-0480-5. [15] J.-S. Guo and Y.-C. Lin, The sign of the wave speed for the Lotka-Volterra competition-diffusion system, Comm. Pure Appl. Anal., 12 (2013), 2083-2090. doi: 10.3934/cpaa.2013.12.2083. [16] J.-S. Guo and C.-H. Wu, Recent developments on wave propagation in 2-species competition systems, Discrete Contin. Dynam. Syst.-B, 17 (2012), 2713-2724. doi: 10.3934/dcdsb.2012.17.2713. [17] J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533. doi: 10.1016/j.jde.2010.12.004. [18] F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in ℝN with conicalshaped level sets, Comm. Partial Differential Equations, 25 (2000), 769-819. doi: 10.1080/03605300008821532. [19] F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ecole Norm. Sup., 37 (2004), 469-506. doi: 10.1016/j.ansens.2004.03.001. [20] F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dynam. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069. [21] F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dynam. Syst., 14 (2006), 75-92. [22] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in ℝN, Arch. Rational Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238. [23] F. Hamel and J.-M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Discrete Contin. Dynam. Syst.-S, 4 (2011), 101-123. doi: 10.3934/dcdss.2011.4.101. [24] M. Haragus and A. Scheel, A bifurcation approach to non-planar traveling waves in reactiondiffusion systems, GAMM-Mitt., 30 (2007), 75-95. doi: 10.1002/gamm.200790012. [25] M. Haragus and A. Scheel, Almost planar waves in anisotropic media, Comm. Partial Differential Equations, 31 (2006), 791-815. doi: 10.1080/03605300500361420. [26] M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré Anal. Linéaire, 23 (2006), 283-329. doi: 10.1016/j.anihpc.2005.03.003. [27] R. Huang, Stability of travelling fronts of the Fisher-KPP equation in ℝN, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 599-622. doi: 10.1007/s00030-008-7041-0. [28] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556. [29] Y. Kan-on, Existence of standing waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 117-133. doi: 10.1007/BF03167302. [30] Y. Kan-on, Instability of stationary solutions for a Lotka-Volterra competition model with diffusion, J. Math. Anal. Appl., 208 (1997), 158-170. doi: 10.1006/jmaa.1997.5309. [31] Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343-349. doi: 10.1007/BF03167252. [32] Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal travelling fronts in the AllenCahn equations, Proc. Royal Soc. Edinburgh Sect. A: Math., 14 (2011), 1031-1054. doi: 10.1017/S0308210510001253. [33] W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reactiondiffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003. [34] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. [35] G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513. doi: 10.1016/j.jde.2007.10.019. [36] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. [37] Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sinica, 3 (2008), 567-584. [38] Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240. doi: 10.1137/080723715. [39] 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. [40] 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. [41] H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dynam. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819. [42] M. del Pino, M. Kowalczyk and J. Wei, A counterexample to a conjecture by De Giorgi in large dimensions, C. R. Math. Acad. Sci. Paris, 346 (2008), 1261-1266. doi: 10.1016/j.crma.2008.10.010. [43] 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. doi: 10.1002/cpa.21438. [44] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, PrenticeHall, Inc. , Englewood Cliffs, N. J. , 1967. 1144 [45] W.-J. Sheng, W.-T. Li and Z.-C. Wang, Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity, J. Differential Equations, 252 (2012), 2388-2424. doi: 10.1016/j.jde.2011.09.01. [46] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, Amer. Math. Soc. , Providence, RI, 1995. [47] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788. [48] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Euqations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037. [49] M. Taniguchi, Multi-Dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dynam. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011. [50] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [51] A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc. , Providence, RI, 1994. [52] Z.-C. 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