American Institute of Mathematical Sciences

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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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

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