American Institute of Mathematical Sciences

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May  2018, 38(5): 2251-2286. doi: 10.3934/dcds.2018093

Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations

Zhen-Hui Bu and  Zhi-Cheng Wang ,

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

* Corresponding author: Z.-C. Wang

Received  April 2017 Revised  December 2017 Published  March 2018

This paper is concerned with the global stability of V-shaped traveling fronts in reaction-diffusion equations with combustion and degenerate monostable nonlinearity. The existence of such curved fronts has been recently proved by [39]. In this paper, by constructing new subsolutions, we show the asymptotic stability of V-shaped traveling fronts.

Keywords: V-shaped traveling front, combustion nonlinearity, degenerate monostable nonlinearity, stability, reaction-diffusion equation.
Mathematics Subject Classification: Primary: 35B35, 35K57; Secondary: 35K55.
Citation: Zhen-Hui Bu, Zhi-Cheng Wang. Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2251-2286. doi: 10.3934/dcds.2018093
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]

P. K. Brazhnik, Exact solutions for the kinematic model of autowaves in two-dimensional excitable media, Physica D, 94 (1996), 205-220. doi: 10.1016/0167-2789(96)00042-5. Google Scholar

[4]

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

[5]

Z.-H. Bu and Z.-C. Wang, Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations Ⅰ, Discrete Contin. Dyn. Syst., 37 (2017), 2395-2430. doi: 10.3934/dcds.2017104. Google Scholar

[6]

C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343. doi: 10.1512/iumj.1984.33.33018. Google Scholar

[7]

M. El Smaily, F. 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

[8]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag, Berlin-New York, 1979. doi: 10.1007/978-3-642-93111-6. Google Scholar

[9]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling wave solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432. Google Scholar

[10]

P. C. Fife and J. B. McLeod, A phase discussion of convergence to traveling fronts for nonlinear diffusions, Arch. Rational Mech. Anal., 75 (1980/81), 281-314. doi: 10.1007/BF00256381. Google Scholar

[11]

R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. Google Scholar

[12]

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

[13]

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

[14]

F. Hamel, R. 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

[15]

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. Google Scholar

[16]

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. doi: 10.3934/dcds.2006.14.75. Google Scholar

[17]

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

[18]

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

[19]

A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Étude de I'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh., 1 (1937), 1-25. Google Scholar

[20]

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

[21]

J. A. Leach, D. 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

[22]

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. Google Scholar

[23]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reactiondiffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. Google Scholar

[24]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. Google Scholar

[25]

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

[26]

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: 10.3934/dcds.2006.15.819. Google Scholar

[27]

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

[28]

H. Ninomiya and M. Taniguchi, Stability of traveling curved fronts in a curvature flow with driving force, Methods Appl. Anal., 8 (2001), 429-450. doi: 10.4310/MAA.2001.v8.n3.a4. Google Scholar

[29]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Insiana Univ. Math. J., 21 (1972), 979-1000. doi: 10.1512/iumj.1972.21.21079. Google Scholar

[30]

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.2011.09.016. Google Scholar

[31]

W.-J. Sheng, W.-T. Li and Z.-C. Wang, Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation, Sci. China Math., 56 (2013), 1969-1982. doi: 10.1007/s11425-013-4699-5. Google Scholar

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

Z.-C. Wang, W.-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

[41]

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), 1869-1908. doi: 10.1007/s11425-016-0015-x. Google Scholar

[42]

Z.-C. Wang, H.-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

[43]

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

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]

P. K. Brazhnik, Exact solutions for the kinematic model of autowaves in two-dimensional excitable media, Physica D, 94 (1996), 205-220. doi: 10.1016/0167-2789(96)00042-5. Google Scholar

[4]

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

[5]

Z.-H. Bu and Z.-C. Wang, Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations Ⅰ, Discrete Contin. Dyn. Syst., 37 (2017), 2395-2430. doi: 10.3934/dcds.2017104. Google Scholar

[6]

C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343. doi: 10.1512/iumj.1984.33.33018. Google Scholar

[7]

M. El Smaily, F. 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

[8]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag, Berlin-New York, 1979. doi: 10.1007/978-3-642-93111-6. Google Scholar

[9]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling wave solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432. Google Scholar

[10]

P. C. Fife and J. B. McLeod, A phase discussion of convergence to traveling fronts for nonlinear diffusions, Arch. Rational Mech. Anal., 75 (1980/81), 281-314. doi: 10.1007/BF00256381. Google Scholar

[11]

R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. Google Scholar

[12]

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

[13]

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

[14]

F. Hamel, R. 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

[15]

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. Google Scholar

[16]

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. doi: 10.3934/dcds.2006.14.75. Google Scholar

[17]

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

[18]

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

[19]

A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Étude de I'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh., 1 (1937), 1-25. Google Scholar

[20]

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

[21]

J. A. Leach, D. 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

[22]

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. Google Scholar

[23]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reactiondiffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. Google Scholar

[24]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. Google Scholar

[25]

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

[26]

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: 10.3934/dcds.2006.15.819. Google Scholar

[27]

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

[28]

H. Ninomiya and M. Taniguchi, Stability of traveling curved fronts in a curvature flow with driving force, Methods Appl. Anal., 8 (2001), 429-450. doi: 10.4310/MAA.2001.v8.n3.a4. Google Scholar

[29]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Insiana Univ. Math. J., 21 (1972), 979-1000. doi: 10.1512/iumj.1972.21.21079. Google Scholar

[30]

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.2011.09.016. Google Scholar

[31]

W.-J. Sheng, W.-T. Li and Z.-C. Wang, Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation, Sci. China Math., 56 (2013), 1969-1982. doi: 10.1007/s11425-013-4699-5. Google Scholar

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

Z.-C. Wang, W.-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

[41]

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), 1869-1908. doi: 10.1007/s11425-016-0015-x. Google Scholar

[42]

Z.-C. Wang, H.-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

[43]

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

Figure 1.  The profiles of the traveling curved front $V$ and the thick solid curve shows the level line $V = 0.5$.
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Figure 2.  The level sets of the traveling curved front $V$.
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