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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May
2018, 38(5): 2251-2286.
doi: 10.3934/dcds.2018093
Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
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 [
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 and Continuous Dynamical Systems,
2018, 38
(5)
: 2251-2286.
doi: 10.3934/dcds.2018093
![]()
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The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.
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Bistable transition fronts in $ \mathbb{R}^{N}$, Adv. Math., 289 (2016), 279-344.
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The evolution of reaction-diffusion waves in a class of scalar reaction-diffusion equations: Algebraic decay rates, Phys. D, 167 (2002), 153-182.
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Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.
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Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379-395.
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H. Ninomiya and M. Taniguchi,
Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832.
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| [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.
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H. Ninomiya and M. Taniguchi,
Stability of traveling curved fronts in a curvature flow with driving force, Methods Appl. Anal., 8 (2001), 429-450.
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D. H. Sattinger,
Monotone methods in nonlinear elliptic and parabolic boundary value problems, Insiana Univ. Math. J., 21 (1972), 979-1000.
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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. |
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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. |
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M. Taniguchi,
Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: 10.1137/060661788. |
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M. Taniguchi,
The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130.
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M. Taniguchi,
Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contnu. Dyn. Syst., 32 (2012), 1011-1046.
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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.
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A. I. Volpert, V. A. Volpert and V. A. Volpert,
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Z.-C. Wang,
Traveling curved fronts in monotone bistable systems, Discrete Contin. Dyn. Syst., 32 (2012), 2339-2374.
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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. |
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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. |
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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.
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| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
R. Fisher,
The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.
|
| [12] |
F. Hamel,
Bistable transition fronts in $ \mathbb{R}^{N}$, Adv. Math., 289 (2016), 279-344.
doi: 10.1016/j.aim.2015.11.033. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [24] |
J. D. Murray,
Mathematical Biology, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [36] |
A. I. Volpert, V. A. Volpert and V. A. Volpert,
Traveling Wave Solutions of Parabolic Systems, 140, Amer. Math. Soc., Providence, RI, 1994. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
Figure 2.
The level sets of the traveling curved front $V$ .
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