# American Institute of Mathematical Sciences

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

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

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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.
Citation: Yan Zhang. Asymptotic behavior of a nonlocal KPP equation with an almost periodic nonlinearity. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, in International conference on differential equations, vol. 1, World Scientific, (2000), 600-605. [18] H. Ishii and S. Koike, Viscosity solutions of functional differential equations, Advances in Mathematical Sciences and Applications, 3 (1994), 191-218. [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. [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. [21] P.-L. Lions, G. Papanicolaou and S. R. Varadhan, Homogenization of Hamilton-Jacobi equations,, Preliminary version., (). [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, in International conference on differential equations, vol. 1, World Scientific, (2000), 600-605. [18] H. Ishii and S. Koike, Viscosity solutions of functional differential equations, Advances in Mathematical Sciences and Applications, 3 (1994), 191-218. [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. [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. [21] P.-L. Lions, G. Papanicolaou and S. R. Varadhan, Homogenization of Hamilton-Jacobi equations,, Preliminary version., (). [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. [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.
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