# 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 & Continuous Dynamical Systems - A, 2016, 36 (9) : 5183-5199. doi: 10.3934/dcds.2016025
##### References:
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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, (1996), 293.   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, (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.  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.  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, 446 (1975), 5.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1214/aop/1176992901.  Google Scholar [17] H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations,, in International conference on differential equations, (2000), 600.   Google Scholar [18] H. Ishii and S. Koike, Viscosity solutions of functional differential equations,, Advances in Mathematical Sciences and Applications, 3 (1994), 191.   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.   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).  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.  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.  doi: 10.3934/dcds.2005.13.1235.  Google Scholar
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