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September  2017, 22(7): 3007-3022. doi: 10.3934/dcdsb.2017160

Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations

Hongmei Cheng 1,, and  Rong Yuan 2,

1. 

School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China
2. 

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

* Corresponding author: Hongmei Cheng

Received  April 2016 Revised  April 2017 Published  May 2017

In this paper, we mainly discuss the existence and asymptotic stability of traveling fronts for the nonlocal evolution equations. With the monostable assumption, we obtain that there exists a constant $c^*>0$, such that the equation has no traveling fronts for $0<c<c^*$ and a traveling front for each c ≥ c*. For $c>c^*$, we will further show that the traveling front is globally asymptotic stable and is unique up to translation. If we applied to some differential equations or integro-differential equations, our results recover and/or complement a number of existing ones.

Keywords: Nonlocal evolution equation, existence, asymptotic stability, traveling fronts, monostable, comparison principle.
Mathematics Subject Classification: Primary:35B05, 35B40, 35A05;Secondary:35R99.
Citation: Hongmei Cheng, Rong Yuan. Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 3007-3022. doi: 10.3934/dcdsb.2017160
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Goldstein J, ed. Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, Berlin: Springer-Verlag, 466 (1975), 5-49. 

[2]

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

[3]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[4]

N. F. Britton, Reaction-Diffusion Equations and Their Applications to Biology Academic Press, San Diego, 1986.

[5]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.

[6]

X. Cabré and J. M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722.  doi: 10.1007/s00220-013-1682-5.

[7]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.

[8]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. 

[9]

X. Chen and J. S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153.

[10]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.  doi: 10.1007/s10231-005-0163-7.

[11]

J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.

[12]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.

[13]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.

[14]

J. Fang and X. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.  doi: 10.1088/0951-7715/24/11/002.

[15]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems Springer Verlag, New York, 1979.

[16]

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

[17]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[18]

F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: Steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753.  doi: 10.1088/0951-7715/27/11/2735.

[19]

A. Kolmogorov, I. Petrovskii and N. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskow. Gos. Univ, 1 (1937), 1-26. 

[20]

S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  doi: 10.1016/j.jde.2005.05.004.

[21]

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.

[22]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.

[23]

M. Mei, C. H. Ou and X. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Appl. Math. , 42 (2010), 2762-2790; Erratum: Erratum: Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal. , 44 (2012), 538-540. doi: 10.1137/110850633.

[24]

J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.

[25]

S. Pan, W. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.  doi: 10.1007/s00033-007-7005-y.

[26]

S. Pan, W. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer Science & Business Media, 1983. doi: 10.1007/978-1-4612-5561-1.

[28]

S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991-1011.  doi: 10.1017/S0308210500003590.

[29]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.

[30]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.

[31]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems American Mathematical Soc. , Providence, RI, 1994.

[32]

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

[33]

Z. Wang, W. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.  doi: 10.1007/s10884-008-9103-8.

[34]

Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations Science Press, Beijing, 1990.

[35]

L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. Differential Equations, 197 (2004), 162-196.  doi: 10.1016/S0022-0396(03)00170-0.

[36]

X. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128.  doi: 10.3934/dcdsb.2004.4.1117.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Goldstein J, ed. Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, Berlin: Springer-Verlag, 466 (1975), 5-49. 

[2]

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

[3]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[4]

N. F. Britton, Reaction-Diffusion Equations and Their Applications to Biology Academic Press, San Diego, 1986.

[5]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.

[6]

X. Cabré and J. M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722.  doi: 10.1007/s00220-013-1682-5.

[7]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.

[8]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. 

[9]

X. Chen and J. S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153.

[10]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.  doi: 10.1007/s10231-005-0163-7.

[11]

J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.

[12]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.

[13]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.

[14]

J. Fang and X. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.  doi: 10.1088/0951-7715/24/11/002.

[15]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems Springer Verlag, New York, 1979.

[16]

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

[17]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[18]

F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: Steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753.  doi: 10.1088/0951-7715/27/11/2735.

[19]

A. Kolmogorov, I. Petrovskii and N. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskow. Gos. Univ, 1 (1937), 1-26. 

[20]

S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  doi: 10.1016/j.jde.2005.05.004.

[21]

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.

[22]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.

[23]

M. Mei, C. H. Ou and X. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Appl. Math. , 42 (2010), 2762-2790; Erratum: Erratum: Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal. , 44 (2012), 538-540. doi: 10.1137/110850633.

[24]

J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.

[25]

S. Pan, W. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.  doi: 10.1007/s00033-007-7005-y.

[26]

S. Pan, W. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer Science & Business Media, 1983. doi: 10.1007/978-1-4612-5561-1.

[28]

S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991-1011.  doi: 10.1017/S0308210500003590.

[29]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.

[30]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.

[31]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems American Mathematical Soc. , Providence, RI, 1994.

[32]

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

[33]

Z. Wang, W. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.  doi: 10.1007/s10884-008-9103-8.

[34]

Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations Science Press, Beijing, 1990.

[35]

L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. Differential Equations, 197 (2004), 162-196.  doi: 10.1016/S0022-0396(03)00170-0.

[36]

X. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128.  doi: 10.3934/dcdsb.2004.4.1117.

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