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How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?
1. | CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France |
References:
[1] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33.
doi: 10.1016/0001-8708(78)90130-5. |
[2] |
B. Audoly, H. Berestycki and Y. Pomeau, Réaction diffusion en écoulement stationnaire rapide,, C. R. Acad. Sci. Paris, 328 (2000), 255.
doi: 10.1016/S1287-4620(00)00115-0. |
[3] |
H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Func. Anal., 255 (2008), 2146.
doi: 10.1016/j.jfa.2008.06.030. |
[4] |
H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena,, Comm. Math. Phys., 253 (2005), 451.
doi: 10.1007/s00220-004-1201-9. |
[5] |
H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for kpp type problems. I - periodic framework,, J. Europ. Math. Soc., 7 (2005), 173.
doi: 10.4171/JEMS/26. |
[6] |
H. Berestycki, F. Hamel and L.Roques, Analysis of the periodically fragmented environment model: II - Biological invasions and pulsating travelling fronts,, J. Math. Pures Appl., 84 (2005), 1101.
doi: 10.1016/j.matpur.2004.10.006. |
[7] |
H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains,, Ann. Mat. Pura Appl., 186 (2007), 469.
doi: 10.1007/s10231-006-0015-0. |
[8] |
H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations,, J. Math. Phys., 53 (2012).
doi: 10.1063/1.4764932. |
[9] |
H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains,, Comm. Pure Appl. Math., 68 (2014), 1014.
doi: 10.1002/cpa.21536. |
[10] |
A. Ducrot, T. Giletti and H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations,, Trans. Amer. Math. Soc., 366 (2014), 5541.
doi: 10.1090/S0002-9947-2014-06105-9. |
[11] |
M. ElSmaily, The non-monotonicity of the KPP speed with respect to diffusion in the presence of a shear flow,, Proceedings of the American Math. Society, 141 (2013), 3553.
doi: 10.1090/S0002-9939-2013-11728-4. |
[12] |
M. ElSmaily and S. Kirsch, The speed of propagation for KPP reaction-diffusion equations within large drift,, Advances in Diff. Equations, 16 (2011), 361.
|
[13] |
R. A. Fisher, The advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335. Google Scholar |
[14] |
M. Freidlin, On wave front propagation in periodic media,, in Stochastic Analysis and Applications (ed. M. Pinsky), (1984), 147. Google Scholar |
[15] |
M. Freidlin, Functional Integration and Partial Differential Equations,, Ann. Math. Stud., (1985).
|
[16] |
M. Freidlin and J. Gartner, On the propagation of concentration waves in periodic and random media,, Sov. Math. Dokl., 249 (1979), 521.
|
[17] |
F. Hamel, G. Nadin and L. Roques, A viscosity solution method for the spreading speed formula in slowly varying media,, Indiana Univ. Math. J., 60 (2011), 1229.
doi: 10.1512/iumj.2011.60.4370. |
[18] |
C. J. Holland, A minimum principle for the principal eigenvalue for second order linear elliptic equation with natural boundary condition,, Comm. Pure Appl. Math., 31 (1978), 509.
doi: 10.1002/cpa.3160310406. |
[19] |
A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 309.
doi: 10.1016/S0294-1449(01)00068-3. |
[20] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Université d'Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), (1937), 1. Google Scholar |
[21] |
X. Liang, X. Lin and H. Matano, Maximizing the spreading speed of KPP fronts in two-dimensional stratified media,, Trans. Amer. Math. Soc., 362 (2010), 5605.
doi: 10.1090/S0002-9947-2010-04931-1. |
[22] |
P.-L. Lions and P. E. Souganidis, Homogenization of "viscous'' Hamilton-Jacobi equations in stationary ergodic media,, Comm. Partial Differential Equations, 30 (2005), 335.
doi: 10.1081/PDE-200050077. |
[23] |
G. Nadin, The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator,, SIAM J. Math. Anal., 41 (2009), 2388.
doi: 10.1137/080743597. |
[24] |
G. Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation,, Eur. J. Appl. Math., 22 (2011), 169.
doi: 10.1017/S0956792511000027. |
[25] |
J. Nolen, A central limit theorem for pulled fronts in a random medium,, Networks and Heterogeneous Media, 6 (2011), 167.
doi: 10.3934/nhm.2011.6.167. |
[26] |
J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional random medium,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1021.
doi: 10.1016/j.anihpc.2009.02.003. |
[27] |
J. Nolen and J. Xin, Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows,, Ann. de l'Inst. Henri Poincare - Analyse Non Lineaire, 26 (2008), 815.
doi: 10.1016/j.anihpc.2008.02.005. |
[28] |
J. Nolen and J. Xin, KPP fronts in 1D random drift,, Discrete and Continuous Dynamical Systems B, 11 (2009), 421.
doi: 10.3934/dcdsb.2009.11.421. |
[29] |
J. Nolen and J. Xin, Variational principle of KPP front speeds in temporally random shear flows with applications,, Communications in Mathematical Physics, 269 (2007), 493.
doi: 10.1007/s00220-006-0144-8. |
[30] |
G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients,, in Proceedings of Conference on Random Fields, (1979), 835.
|
[31] |
L. Ryzhik and A. Zlatos, KPP pulsating front speed-up by flows,, Commun. Math. Sci., 5 (2007), 575.
doi: 10.4310/CMS.2007.v5.n3.a4. |
[32] |
N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments,, Theor. Population Biol., 30 (1986), 143.
doi: 10.1016/0040-5809(86)90029-8. |
[33] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice,, Oxford Series in Ecology and Evolution, (1997). Google Scholar |
[34] |
P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications,, Asymptotic Analysis, 20 (1999), 1.
|
[35] |
V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994).
doi: 10.1007/978-3-642-84659-5. |
[36] |
A. Zlatos, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows,, Arch. Ration. Mech. Anal., 195 (2009), 441.
doi: 10.1007/s00205-009-0282-1. |
show all references
References:
[1] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33.
doi: 10.1016/0001-8708(78)90130-5. |
[2] |
B. Audoly, H. Berestycki and Y. Pomeau, Réaction diffusion en écoulement stationnaire rapide,, C. R. Acad. Sci. Paris, 328 (2000), 255.
doi: 10.1016/S1287-4620(00)00115-0. |
[3] |
H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Func. Anal., 255 (2008), 2146.
doi: 10.1016/j.jfa.2008.06.030. |
[4] |
H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena,, Comm. Math. Phys., 253 (2005), 451.
doi: 10.1007/s00220-004-1201-9. |
[5] |
H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for kpp type problems. I - periodic framework,, J. Europ. Math. Soc., 7 (2005), 173.
doi: 10.4171/JEMS/26. |
[6] |
H. Berestycki, F. Hamel and L.Roques, Analysis of the periodically fragmented environment model: II - Biological invasions and pulsating travelling fronts,, J. Math. Pures Appl., 84 (2005), 1101.
doi: 10.1016/j.matpur.2004.10.006. |
[7] |
H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains,, Ann. Mat. Pura Appl., 186 (2007), 469.
doi: 10.1007/s10231-006-0015-0. |
[8] |
H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations,, J. Math. Phys., 53 (2012).
doi: 10.1063/1.4764932. |
[9] |
H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains,, Comm. Pure Appl. Math., 68 (2014), 1014.
doi: 10.1002/cpa.21536. |
[10] |
A. Ducrot, T. Giletti and H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations,, Trans. Amer. Math. Soc., 366 (2014), 5541.
doi: 10.1090/S0002-9947-2014-06105-9. |
[11] |
M. ElSmaily, The non-monotonicity of the KPP speed with respect to diffusion in the presence of a shear flow,, Proceedings of the American Math. Society, 141 (2013), 3553.
doi: 10.1090/S0002-9939-2013-11728-4. |
[12] |
M. ElSmaily and S. Kirsch, The speed of propagation for KPP reaction-diffusion equations within large drift,, Advances in Diff. Equations, 16 (2011), 361.
|
[13] |
R. A. Fisher, The advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335. Google Scholar |
[14] |
M. Freidlin, On wave front propagation in periodic media,, in Stochastic Analysis and Applications (ed. M. Pinsky), (1984), 147. Google Scholar |
[15] |
M. Freidlin, Functional Integration and Partial Differential Equations,, Ann. Math. Stud., (1985).
|
[16] |
M. Freidlin and J. Gartner, On the propagation of concentration waves in periodic and random media,, Sov. Math. Dokl., 249 (1979), 521.
|
[17] |
F. Hamel, G. Nadin and L. Roques, A viscosity solution method for the spreading speed formula in slowly varying media,, Indiana Univ. Math. J., 60 (2011), 1229.
doi: 10.1512/iumj.2011.60.4370. |
[18] |
C. J. Holland, A minimum principle for the principal eigenvalue for second order linear elliptic equation with natural boundary condition,, Comm. Pure Appl. Math., 31 (1978), 509.
doi: 10.1002/cpa.3160310406. |
[19] |
A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 309.
doi: 10.1016/S0294-1449(01)00068-3. |
[20] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Université d'Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), (1937), 1. Google Scholar |
[21] |
X. Liang, X. Lin and H. Matano, Maximizing the spreading speed of KPP fronts in two-dimensional stratified media,, Trans. Amer. Math. Soc., 362 (2010), 5605.
doi: 10.1090/S0002-9947-2010-04931-1. |
[22] |
P.-L. Lions and P. E. Souganidis, Homogenization of "viscous'' Hamilton-Jacobi equations in stationary ergodic media,, Comm. Partial Differential Equations, 30 (2005), 335.
doi: 10.1081/PDE-200050077. |
[23] |
G. Nadin, The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator,, SIAM J. Math. Anal., 41 (2009), 2388.
doi: 10.1137/080743597. |
[24] |
G. Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation,, Eur. J. Appl. Math., 22 (2011), 169.
doi: 10.1017/S0956792511000027. |
[25] |
J. Nolen, A central limit theorem for pulled fronts in a random medium,, Networks and Heterogeneous Media, 6 (2011), 167.
doi: 10.3934/nhm.2011.6.167. |
[26] |
J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional random medium,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1021.
doi: 10.1016/j.anihpc.2009.02.003. |
[27] |
J. Nolen and J. Xin, Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows,, Ann. de l'Inst. Henri Poincare - Analyse Non Lineaire, 26 (2008), 815.
doi: 10.1016/j.anihpc.2008.02.005. |
[28] |
J. Nolen and J. Xin, KPP fronts in 1D random drift,, Discrete and Continuous Dynamical Systems B, 11 (2009), 421.
doi: 10.3934/dcdsb.2009.11.421. |
[29] |
J. Nolen and J. Xin, Variational principle of KPP front speeds in temporally random shear flows with applications,, Communications in Mathematical Physics, 269 (2007), 493.
doi: 10.1007/s00220-006-0144-8. |
[30] |
G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients,, in Proceedings of Conference on Random Fields, (1979), 835.
|
[31] |
L. Ryzhik and A. Zlatos, KPP pulsating front speed-up by flows,, Commun. Math. Sci., 5 (2007), 575.
doi: 10.4310/CMS.2007.v5.n3.a4. |
[32] |
N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments,, Theor. Population Biol., 30 (1986), 143.
doi: 10.1016/0040-5809(86)90029-8. |
[33] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice,, Oxford Series in Ecology and Evolution, (1997). Google Scholar |
[34] |
P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications,, Asymptotic Analysis, 20 (1999), 1.
|
[35] |
V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994).
doi: 10.1007/978-3-642-84659-5. |
[36] |
A. Zlatos, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows,, Arch. Ration. Mech. Anal., 195 (2009), 441.
doi: 10.1007/s00205-009-0282-1. |
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