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Asymptotic behavior of a nonlocal KPP equation with an almost periodic nonlinearity
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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.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
M. Freidlin, Limit theorems for large deviations and reaction-diffusion equations,, The Annals of Probability, 13 (1985), 639.
doi: 10.1214/aop/1176992901. |
[17] |
H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations,, in International conference on differential equations, (2000), 600.
|
[18] |
H. Ishii and S. Koike, Viscosity solutions of functional differential equations,, Advances in Mathematical Sciences and Applications, 3 (1994), 191.
|
[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. |
[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. |
[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. |
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, (1996), 293.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
M. Freidlin, Limit theorems for large deviations and reaction-diffusion equations,, The Annals of Probability, 13 (1985), 639.
doi: 10.1214/aop/1176992901. |
[17] |
H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations,, in International conference on differential equations, (2000), 600.
|
[18] |
H. Ishii and S. Koike, Viscosity solutions of functional differential equations,, Advances in Mathematical Sciences and Applications, 3 (1994), 191.
|
[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. |
[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. |
[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. |
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