May  2017, 37(5): 2395-2430. doi: 10.3934/dcds.2017104

Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations Ⅰ

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author: Z.-C. Wang

Received  July 2016 Revised  December 2016 Published  February 2017

This paper is concerned with traveling curved fronts in reaction diffusion equations with degenerate monostable and combustion nonlinearities. For a given admissible pyramidal in three-dimensional spaces, the existence of a pyramidal traveling front has been proved by Wang and Bu [30] recently. By constructing new supersolutions and developing the arguments of Taniguchi [25] for the Allen-Cahn equation, in this paper we first characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces, and then establish the uniqueness and asymptotic stability of such three-dimensional pyramidal traveling fronts under the condition that given perturbations decay at infinity.

Citation: Zhen-Hui Bu, Zhi-Cheng Wang. Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations Ⅰ. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2395-2430. doi: 10.3934/dcds.2017104
References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. Google Scholar

[2]

A. Bonnet and F. Hamel, Existence of non-planar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80-118. doi: 10.1137/s0036141097316391. Google Scholar

[3]

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. Google Scholar

[4]

Z. -H. Bu and Z. -C. Wang, Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations, submitted.Google Scholar

[5]

Z. -H. Bu and Z. -C. Wang, Stability of pyramidal traveling fronts in degenerate monostable and combustion equations Ⅱ, preprint.Google Scholar

[6]

M. El SmailyF. Hamel and R. Huang, Two-dimensional curved fronts in a periodic shear flow, Nonlinear Anal., 74 (2011), 6469-6486. doi: 10.1016/j.na.2011.06.030. Google Scholar

[7] G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. Google Scholar
[8]

F. Hamel, Bistable transition fronts in $\mathbb{R}^{N}$, Adv. Math., 289 (2016), 279-344. doi: 10.1016/j.aim.2015.11.033. Google Scholar

[9]

F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in $\mathbb{R}^{N}$ with conicalshaped level sets, Comm. Partial Differential Equations, 25 (2000), 769-819. doi: 10.1080/03605300008821532. Google Scholar

[10]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238. Google Scholar

[11]

F. HamelR. Monneau and J.-M. Roquejoffre, Stability of conical fronts in a model for conical flames in two space dimensions, Ann. Sci. École Normale Sup., 37 (2004), 469-506. doi: 10.1016/j.ansens.2004.03.001. Google Scholar

[12]

F. HamelR. 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. Google Scholar

[13]

F. HamelR. Monneau and J.-M Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92. doi: 10.3934/dcds.2006.14.75. Google Scholar

[14]

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. Google Scholar

[15]

M. Haragus and A. Scheel, Almost planar waves in anisotropic media, Comm. Partial Differential Equations, 31 (2006), 791-815. doi: 10.1080/03605300500361420. Google Scholar

[16]

R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $\mathbb{R}^{N}$, Nonlinear Diff. Eq. Appl., 15 (2008), 599-622. doi: 10.1007/s00030-008-7041-0. Google Scholar

[17]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253. Google Scholar

[18]

J. A. LeachD. J. Needham and A. L. Kay, The evolution of reaction-diffusion waves in a class of scalar reaction-diffusion equations: Algebraic decay rates, Phys. D, 167 (2002), 153-182. doi: 10.1016/S0167-2789(02)00428-1. Google Scholar

[19]

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. Google Scholar

[20]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832. doi: 10.1016/j.jde.2004.06.011. Google Scholar

[21]

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. Google Scholar

[22]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Insiana Univ. Math. J., 21 (1972), 979-1000. Google Scholar

[23]

W.-J. ShengW.-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.2011.09.016. Google Scholar

[24]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788. Google Scholar

[25]

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. Google Scholar

[26]

M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contnu. Dyn. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011. Google Scholar

[27]

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. Google Scholar

[28]

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. doi: 10.1016/j.jde.2015.11.010. Google Scholar

[29]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems 140, Amer. Math. Soc. , Providence, RI, 1994. Google Scholar

[30]

Z.-C. Wang and Z.-H. Bu, Nonplanar traveling fronts in reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearity, J. Differential Equations, 260 (2016), 6405-6450. doi: 10.1016/j.jde.2015.12.045. Google Scholar

[31]

Z.-C. WangW.-T. Li and S. Ruan, Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems, Sci. China Math., 59 (2016), 1869-1908. doi: 10.1007/s11425-016-0015-x. Google Scholar

[32]

Z.-C. WangW.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advecion diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025. Google Scholar

[33]

Z.-C. WangH.-L. Niu and S. Ruan, On the existence of axisymmetric traveling fronts in the Lotka-Volterra competition-diffusion system in $\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst -B, 22 (2017), 1111-1144. doi: 10.3934/dcdsb.2017055. Google Scholar

[34]

Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196-3229. doi: 10.1016/j.jde.2011.01.017. Google Scholar

[35]

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. Google Scholar

[36]

Z.-C. Wang, Cylindrically symmetric traveling fronts in reaction-diffusion equations with bistable nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1053-1090. doi: 10.1017/S0308210515000268. Google Scholar

[37]

Y.-P. Wu and X.-X. Xing, Stability of traveling waves with critical speeds for p-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 20 (2008), 1123-1139. doi: 10.3934/dcds.2008.20.1123. Google Scholar

[38]

Y.-P. WuX.-X. Xing and Q.-X. Ye, Stability of traveling waves with algebraic decay for n-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 16 (2006), 47-66. doi: 10.3934/dcds.2006.16.47. Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. Google Scholar

[2]

A. Bonnet and F. Hamel, Existence of non-planar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80-118. doi: 10.1137/s0036141097316391. Google Scholar

[3]

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. Google Scholar

[4]

Z. -H. Bu and Z. -C. Wang, Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations, submitted.Google Scholar

[5]

Z. -H. Bu and Z. -C. Wang, Stability of pyramidal traveling fronts in degenerate monostable and combustion equations Ⅱ, preprint.Google Scholar

[6]

M. El SmailyF. Hamel and R. Huang, Two-dimensional curved fronts in a periodic shear flow, Nonlinear Anal., 74 (2011), 6469-6486. doi: 10.1016/j.na.2011.06.030. Google Scholar

[7] G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. Google Scholar
[8]

F. Hamel, Bistable transition fronts in $\mathbb{R}^{N}$, Adv. Math., 289 (2016), 279-344. doi: 10.1016/j.aim.2015.11.033. Google Scholar

[9]

F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in $\mathbb{R}^{N}$ with conicalshaped level sets, Comm. Partial Differential Equations, 25 (2000), 769-819. doi: 10.1080/03605300008821532. Google Scholar

[10]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238. Google Scholar

[11]

F. HamelR. Monneau and J.-M. Roquejoffre, Stability of conical fronts in a model for conical flames in two space dimensions, Ann. Sci. École Normale Sup., 37 (2004), 469-506. doi: 10.1016/j.ansens.2004.03.001. Google Scholar

[12]

F. HamelR. 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. Google Scholar

[13]

F. HamelR. Monneau and J.-M Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92. doi: 10.3934/dcds.2006.14.75. Google Scholar

[14]

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. Google Scholar

[15]

M. Haragus and A. Scheel, Almost planar waves in anisotropic media, Comm. Partial Differential Equations, 31 (2006), 791-815. doi: 10.1080/03605300500361420. Google Scholar

[16]

R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $\mathbb{R}^{N}$, Nonlinear Diff. Eq. Appl., 15 (2008), 599-622. doi: 10.1007/s00030-008-7041-0. Google Scholar

[17]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253. Google Scholar

[18]

J. A. LeachD. J. Needham and A. L. Kay, The evolution of reaction-diffusion waves in a class of scalar reaction-diffusion equations: Algebraic decay rates, Phys. D, 167 (2002), 153-182. doi: 10.1016/S0167-2789(02)00428-1. Google Scholar

[19]

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. Google Scholar

[20]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832. doi: 10.1016/j.jde.2004.06.011. Google Scholar

[21]

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. Google Scholar

[22]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Insiana Univ. Math. J., 21 (1972), 979-1000. Google Scholar

[23]

W.-J. ShengW.-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.2011.09.016. Google Scholar

[24]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788. Google Scholar

[25]

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. Google Scholar

[26]

M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contnu. Dyn. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011. Google Scholar

[27]

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. Google Scholar

[28]

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. doi: 10.1016/j.jde.2015.11.010. Google Scholar

[29]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems 140, Amer. Math. Soc. , Providence, RI, 1994. Google Scholar

[30]

Z.-C. Wang and Z.-H. Bu, Nonplanar traveling fronts in reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearity, J. Differential Equations, 260 (2016), 6405-6450. doi: 10.1016/j.jde.2015.12.045. Google Scholar

[31]

Z.-C. WangW.-T. Li and S. Ruan, Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems, Sci. China Math., 59 (2016), 1869-1908. doi: 10.1007/s11425-016-0015-x. Google Scholar

[32]

Z.-C. WangW.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advecion diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025. Google Scholar

[33]

Z.-C. WangH.-L. Niu and S. Ruan, On the existence of axisymmetric traveling fronts in the Lotka-Volterra competition-diffusion system in $\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst -B, 22 (2017), 1111-1144. doi: 10.3934/dcdsb.2017055. Google Scholar

[34]

Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196-3229. doi: 10.1016/j.jde.2011.01.017. Google Scholar

[35]

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. Google Scholar

[36]

Z.-C. Wang, Cylindrically symmetric traveling fronts in reaction-diffusion equations with bistable nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1053-1090. doi: 10.1017/S0308210515000268. Google Scholar

[37]

Y.-P. Wu and X.-X. Xing, Stability of traveling waves with critical speeds for p-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 20 (2008), 1123-1139. doi: 10.3934/dcds.2008.20.1123. Google Scholar

[38]

Y.-P. WuX.-X. Xing and Q.-X. Ye, Stability of traveling waves with algebraic decay for n-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 16 (2006), 47-66. doi: 10.3934/dcds.2006.16.47. Google Scholar

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