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September  2016, 36(9): 5183-5199. doi: 10.3934/dcds.2016025

Asymptotic behavior of a nonlocal KPP equation with an almost periodic nonlinearity

Yan Zhang 1,

1. 

35 River Drive S. Apt. 210, Jersey City, NJ 07310, United States

Received  December 2014 Revised  February 2016 Published  May 2016

We consider a space-inhomogeneous KPP equation with a nonlocal diffusion and an almost-periodic nonlinearity. By employing and adapting the theory of homogenization, we show that solutions of this equation asymptotically converge to its stationary states in regions of space separated by a front that is determined by a Hamilton-Jacobi variational inequality.
Keywords: partial integro-differential equation, Asymptotic behavior, viscosity solutions., reaction-diffusion equation, homogenization.
Mathematics Subject Classification: Primary: 35R09, 35K57; Secondary: 35B27, 35B4.
Citation: Yan Zhang. Asymptotic behavior of a nonlocal KPP equation with an almost periodic nonlinearity. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5183-5199. doi: 10.3934/dcds.2016025
References:
[1]

O. Alvarez and A. Tourin, Viscosity solutions of nonlinear integro-differential equations, in Annales de l'Institut Henri Poincaré. Analyse non liné}aire, vol. 13, Elsevier, 1996, 293-317.  Google Scholar

[2]

M. Arisawa, Viscosity solution's approach to jump processes arising in mathematical finances, in Proceedings of 10th International conference on mathematical finances sponcered by Daiwa securuty insurance, Dep. of Economics in Kyoto U, 2005. Google Scholar

[3]

M. Arisawa, A localization of the Lévy operators arising in mathematical finances, Stochastic Processes and Applications To Mathematical Finance, (2007), 23-52, arXiv:1012.3108 doi: 10.1142/9789812770448_0002.  Google Scholar

[4]

M. Arisawa, Homogenization of a class of integro-differential equations with Lévy operators, Communications in Partial Differential Equations, 34 (2009), 617-624. arXiv:1012.4163 doi: 10.1080/03605300902963518.  Google Scholar

[5]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial differential equations and related topics, Springer, 446 (1975), 5-49.  Google Scholar

[6]

G. Barles, E. Chasseigne and C. Imbert et al., On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana University Mathematics Journal, 57 (2008), 213-246. doi: 10.1512/iumj.2008.57.3315.  Google Scholar

[7]

G. Barles, S. Mirrahimi and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equations: A general convergence result, Methods and Applications of Analysis, 16 (2009), 321-340. arXiv:0903.4952. doi: 10.4310/MAA.2009.v16.n3.a4.  Google Scholar

[8]

H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-{KPP} equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[9]

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

[10]

J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 30 (2013), 179-223. doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[11]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[12]

M. G. Crandall, P.-L. Lions and P. E. Souganidis, Maximal solutions and universal bounds for some partial differential equations of evolution, Archive for Rational Mechanics and Analysis, 105 (1989), 163-190. doi: 10.1007/BF00250835.  Google Scholar

[13]

L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. doi: 10.1017/S0308210500032121.  Google Scholar

[14]

L. Evans and P. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana University mathematics journal, 38 (1989), 141-172. doi: 10.1512/iumj.1989.38.38007.  Google Scholar

[15]

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

[16]

M. Freidlin, Limit theorems for large deviations and reaction-diffusion equations, The Annals of Probability, 13 (1985), 639-675. doi: 10.1214/aop/1176992901.  Google Scholar

[17]

H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, in International conference on differential equations, vol. 1, World Scientific, (2000), 600-605.  Google Scholar

[18]

H. Ishii and S. Koike, Viscosity solutions of functional differential equations, Advances in Mathematical Sciences and Applications, 3 (1994), 191-218.  Google Scholar

[19]

A. Kolmogorov, I. Petrovsky and N. Piskunov, Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem, Bull. Moscow State Univ. Ser. A: Math. Mech, 1 (1937), 1-25. Google Scholar

[20]

T. S. Lim and A. Zlatos, Transition fronts for inhomogeneous Fisher-{KPP} reactions and non-local diffusion, Trans. Amer. Math. Soc., 2015, arXiv:1403.0166. doi: 10.1090/tran/6602.  Google Scholar

[21]

P.-L. Lions, G. Papanicolaou and S. R. Varadhan, Homogenization of Hamilton-Jacobi equations,, Preliminary version., ().   Google Scholar

[22]

A. J. Majda and P. E. Souganidis, Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7 (1994), 1-30. doi: 10.1088/0951-7715/7/1/001.  Google Scholar

[23]

B. Perthame and P. Souganidis, Front propagation for a jump process model arising in spatial ecology, Dynamical Systems, 13 (2005), 1235-1246. doi: 10.3934/dcds.2005.13.1235.  Google Scholar

show all references

References:
[1]

O. Alvarez and A. Tourin, Viscosity solutions of nonlinear integro-differential equations, in Annales de l'Institut Henri Poincaré. Analyse non liné}aire, vol. 13, Elsevier, 1996, 293-317.  Google Scholar

[2]

M. Arisawa, Viscosity solution's approach to jump processes arising in mathematical finances, in Proceedings of 10th International conference on mathematical finances sponcered by Daiwa securuty insurance, Dep. of Economics in Kyoto U, 2005. Google Scholar

[3]

M. Arisawa, A localization of the Lévy operators arising in mathematical finances, Stochastic Processes and Applications To Mathematical Finance, (2007), 23-52, arXiv:1012.3108 doi: 10.1142/9789812770448_0002.  Google Scholar

[4]

M. Arisawa, Homogenization of a class of integro-differential equations with Lévy operators, Communications in Partial Differential Equations, 34 (2009), 617-624. arXiv:1012.4163 doi: 10.1080/03605300902963518.  Google Scholar

[5]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial differential equations and related topics, Springer, 446 (1975), 5-49.  Google Scholar

[6]

G. Barles, E. Chasseigne and C. Imbert et al., On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana University Mathematics Journal, 57 (2008), 213-246. doi: 10.1512/iumj.2008.57.3315.  Google Scholar

[7]

G. Barles, S. Mirrahimi and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equations: A general convergence result, Methods and Applications of Analysis, 16 (2009), 321-340. arXiv:0903.4952. doi: 10.4310/MAA.2009.v16.n3.a4.  Google Scholar

[8]

H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-{KPP} equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[9]

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

[10]

J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 30 (2013), 179-223. doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[11]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[12]

M. G. Crandall, P.-L. Lions and P. E. Souganidis, Maximal solutions and universal bounds for some partial differential equations of evolution, Archive for Rational Mechanics and Analysis, 105 (1989), 163-190. doi: 10.1007/BF00250835.  Google Scholar

[13]

L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. doi: 10.1017/S0308210500032121.  Google Scholar

[14]

L. Evans and P. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana University mathematics journal, 38 (1989), 141-172. doi: 10.1512/iumj.1989.38.38007.  Google Scholar

[15]

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

[16]

M. Freidlin, Limit theorems for large deviations and reaction-diffusion equations, The Annals of Probability, 13 (1985), 639-675. doi: 10.1214/aop/1176992901.  Google Scholar

[17]

H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, in International conference on differential equations, vol. 1, World Scientific, (2000), 600-605.  Google Scholar

[18]

H. Ishii and S. Koike, Viscosity solutions of functional differential equations, Advances in Mathematical Sciences and Applications, 3 (1994), 191-218.  Google Scholar

[19]

A. Kolmogorov, I. Petrovsky and N. Piskunov, Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem, Bull. Moscow State Univ. Ser. A: Math. Mech, 1 (1937), 1-25. Google Scholar

[20]

T. S. Lim and A. Zlatos, Transition fronts for inhomogeneous Fisher-{KPP} reactions and non-local diffusion, Trans. Amer. Math. Soc., 2015, arXiv:1403.0166. doi: 10.1090/tran/6602.  Google Scholar

[21]

P.-L. Lions, G. Papanicolaou and S. R. Varadhan, Homogenization of Hamilton-Jacobi equations,, Preliminary version., ().   Google Scholar

[22]

A. J. Majda and P. E. Souganidis, Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7 (1994), 1-30. doi: 10.1088/0951-7715/7/1/001.  Google Scholar

[23]

B. Perthame and P. Souganidis, Front propagation for a jump process model arising in spatial ecology, Dynamical Systems, 13 (2005), 1235-1246. doi: 10.3934/dcds.2005.13.1235.  Google Scholar

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