June  2012, 32(6): 2339-2374. doi: 10.3934/dcds.2012.32.2339

Traveling curved fronts in monotone bistable systems

1. 

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

Received  January 2011 Revised  October 2011 Published  February 2012

This paper is concerned with the existence, uniqueness and stability of traveling curved fronts for reaction-diffusion bistable systems in two-dimensional space. By establishing the comparison theorem and constructing appropriate supersolutions and subsolutions, we prove the existence of traveling curved fronts. Furthermore, we show that the curved front is globally stable. Finally, we apply the results to three important models in biology.
Citation: Zhi-Cheng Wang. Traveling curved fronts in monotone bistable systems. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2339-2374. doi: 10.3934/dcds.2012.32.2339
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]

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.

[3]

P. K. Brazhnik and J. J. Tyson, On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (2000), 371-391. doi: 10.1137/S0036139997325497.

[4]

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.

[5]

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.

[6]

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.

[7]

M. Feinberg and D. Terman, Traveling composition waves on isothermal catalyst surfaces, Arch. Rational Mech. Anal., 116 (1991), 35-69. doi: 10.1007/BF00375602.

[8]

P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.

[9]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Arch. Rational Mech. Anal., 65 (1977), 355-361.

[10]

P. C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusions,, Arch. Rational Mech. Anal., 75 (): 281. 

[11]

A. Friedman, "Partial Differential Equations Of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[12]

R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretical approach, J. Differential Equations, 44 (1982), 343-364. doi: 10.1016/0022-0396(82)90001-8.

[13]

S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822. doi: 10.1137/S003614100139991.

[14]

C. Gui, Symmetry of travelling wave solutions to the Allen-Cahn equation in $\mathbbR^2$,, preprint, (). 

[15]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. École Norm. Sup. (4), 37 (2004), 469-506.

[16]

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.

[17]

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.

[18]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbbR^N$, Arch. Rational Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238.

[19]

F. Hamel and J.-M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Disc. Cont. Dyn. Systems Ser. S, 4 (2011), 101-123. doi: 10.3934/dcdss.2011.4.101.

[20]

M. Haragus and A. Scheel, A bifurcation approach to non-planar traveling waves in reaction-diffusion systems, GAMM-Mitt., 30 (2007), 75-95. doi: 10.1002/gamm.200790012.

[21]

M. Haragus and A. Scheel, Almost planar waves in anisotropic media, Comm. Partial Differential Equations, 31 (2006), 791-815.

[22]

M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 283-329.

[23]

R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $\mathbbR^N$, Nonlinear Diff. Eq. Appl., 15 (2008), 599-622.

[24]

T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269. doi: 10.1090/S0002-9947-97-01668-1.

[25]

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.

[26]

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.

[27]

C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. II, Comm. Partial Differential Equations, 17 (1992), 1901-1924.

[28]

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.

[29]

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.

[30]

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

[31]

K. Mischaikow and V. Hutson, Travelling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008. doi: 10.1137/0524059.

[32]

Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sinica (N.S.), 3 (2008), 567-584.

[33]

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.

[34]

J. D. Murray, "Mathematical Biology,'' Biomathematics, 19, Springer-Verlag, Berlin, 1989.

[35]

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.

[36]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contnu. Dyn. Syst., 15 (2006), 829-832.

[37]

T. Ogiwara and H. Matano, Monotonicity and convergence results in order-preserving systems in the presence of symmetry, Discrete Contnu. Dyn. Syst., 5 (1999), 1-34.

[38]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.

[39]

J.-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl. (4), 188 (2009), 207-233.

[40]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana University Math. J., 21 (): 979.  doi: 10.1512/iumj.1972.21.21079.

[41]

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

[42]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Euqations, 246 (2009), 2103-2130.

[43]

J.-C. Tsai, Global exponential stability of traveling waves in monotone bistable systems, Discrete Contnu. Dyn. Syst., 21 (2008), 601-623. doi: 10.3934/dcds.2008.21.601.

[44]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, RI, 1994.

[45]

M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity, 23 (2010), 1609-1630. doi: 10.1088/0951-7715/23/7/005.

[46]

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.

[47]

J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I, Comm. Partial Differential Equations, 17 (1992), 1889-1899.

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]

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.

[3]

P. K. Brazhnik and J. J. Tyson, On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (2000), 371-391. doi: 10.1137/S0036139997325497.

[4]

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.

[5]

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.

[6]

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.

[7]

M. Feinberg and D. Terman, Traveling composition waves on isothermal catalyst surfaces, Arch. Rational Mech. Anal., 116 (1991), 35-69. doi: 10.1007/BF00375602.

[8]

P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.

[9]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Arch. Rational Mech. Anal., 65 (1977), 355-361.

[10]

P. C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusions,, Arch. Rational Mech. Anal., 75 (): 281. 

[11]

A. Friedman, "Partial Differential Equations Of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[12]

R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretical approach, J. Differential Equations, 44 (1982), 343-364. doi: 10.1016/0022-0396(82)90001-8.

[13]

S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822. doi: 10.1137/S003614100139991.

[14]

C. Gui, Symmetry of travelling wave solutions to the Allen-Cahn equation in $\mathbbR^2$,, preprint, (). 

[15]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. École Norm. Sup. (4), 37 (2004), 469-506.

[16]

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.

[17]

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.

[18]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbbR^N$, Arch. Rational Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238.

[19]

F. Hamel and J.-M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Disc. Cont. Dyn. Systems Ser. S, 4 (2011), 101-123. doi: 10.3934/dcdss.2011.4.101.

[20]

M. Haragus and A. Scheel, A bifurcation approach to non-planar traveling waves in reaction-diffusion systems, GAMM-Mitt., 30 (2007), 75-95. doi: 10.1002/gamm.200790012.

[21]

M. Haragus and A. Scheel, Almost planar waves in anisotropic media, Comm. Partial Differential Equations, 31 (2006), 791-815.

[22]

M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 283-329.

[23]

R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $\mathbbR^N$, Nonlinear Diff. Eq. Appl., 15 (2008), 599-622.

[24]

T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269. doi: 10.1090/S0002-9947-97-01668-1.

[25]

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.

[26]

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.

[27]

C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. II, Comm. Partial Differential Equations, 17 (1992), 1901-1924.

[28]

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.

[29]

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.

[30]

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

[31]

K. Mischaikow and V. Hutson, Travelling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008. doi: 10.1137/0524059.

[32]

Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sinica (N.S.), 3 (2008), 567-584.

[33]

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.

[34]

J. D. Murray, "Mathematical Biology,'' Biomathematics, 19, Springer-Verlag, Berlin, 1989.

[35]

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.

[36]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contnu. Dyn. Syst., 15 (2006), 829-832.

[37]

T. Ogiwara and H. Matano, Monotonicity and convergence results in order-preserving systems in the presence of symmetry, Discrete Contnu. Dyn. Syst., 5 (1999), 1-34.

[38]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.

[39]

J.-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl. (4), 188 (2009), 207-233.

[40]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana University Math. J., 21 (): 979.  doi: 10.1512/iumj.1972.21.21079.

[41]

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

[42]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Euqations, 246 (2009), 2103-2130.

[43]

J.-C. Tsai, Global exponential stability of traveling waves in monotone bistable systems, Discrete Contnu. Dyn. Syst., 21 (2008), 601-623. doi: 10.3934/dcds.2008.21.601.

[44]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, RI, 1994.

[45]

M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity, 23 (2010), 1609-1630. doi: 10.1088/0951-7715/23/7/005.

[46]

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.

[47]

J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I, Comm. Partial Differential Equations, 17 (1992), 1889-1899.

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