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The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow
1. | Institut de Mathématiques (UMR CNRS 5219), Université Paul Sabatier, 31062 Toulouse Cedex 4, France |
2. | Laboratoire MIP, Université Paul Sabatier, 31062 Toulouse Cedex 9 |
References:
[1] |
B. Audoly, H. Berestycki and Y. Pomeau, Réaction-diffusion en écoulement rapide, C. R. Acad. Sci.Paris, série II, 328 (2000), 255-262. |
[2] |
M. Bages, "Équations de réaction-diffusion de type KPP: ondes pulsatoires, dynamique non triviale et applications," Ph.D thesis, Université de Toulouse, 2007. |
[3] |
M. Bages, P. Martinez and J.-M. Roquejoffre, Dynamique en grand temps pour une classe d'équations de type KPP en milieu périodique, C.R. Acad. Sci. Paris, Ser. I, 346 (2008), 1051-1056. |
[4] |
M. Bages, P. Martinez and J.-M. Roquejoffre, How travelling waves attract solutions of KPP-type equations, Trans A.M.S., to appear. |
[5] |
H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, In "Perspectives in Nonlinear Partial Differential Equations," in honor of H. Brezis, Contemp. Math., 446 Amer. Math. Soc., 101-123. |
[6] |
F. Benkhaldoun and B. Larrouturou, Numerical analysis of the two-dimensional thermodiffusive model for flame propagation, RAIRO Modèl. Math. Anal. Numér., 22 (1988), 535-560. |
[7] |
H. Berestycki and B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, vol. X, H. Brezis & J.-L. Lions eds., Pitman, London, (1991), 65-129. |
[8] |
H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré, Anal. non Linéaire, 9 (1992), 497-572. |
[9] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Memoirs of the AMS, 44 (1983), Providence, R.I. |
[10] |
P. Collet and J. P. Eckmann, Space-time behaviour in problems of hydrodynamic type: a case study, Nonlinearity, 5 (1992), 1265-1302.
doi: 10.1088/0951-7715/5/6/004. |
[11] |
U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Physica D, 146 (2000), 1-99.
doi: 10.1016/S0167-2789(00)00068-3. |
[12] |
A. Fannjiang and G. Papanicolaou, Convection Enhanced Diffusion for Periodic Flows, SIAM J. Appl. Math., 54 (1994), 333-408.
doi: 10.1137/S0036139992236785. |
[13] |
F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\RR^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[14] |
F. Hamel and L. Roques, Uniqueness and stability of monostable pulsating travelling fronts, J. European Math. Soc, to appear. |
[15] |
F. Hamel and L. Ryzhik, Non-adiabatic KPP fronts with an arbitrary Lewis number, Nonlinearity, 18 (2005), 2881-2902.
doi: 10.1088/0951-7715/18/6/024. |
[16] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840,1990, Springer. |
[17] |
A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Ann. de l'Inst. Henri Poincaré, C. Analyse non linéaire, 18 (2001), 309-358. |
[18] |
A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskowskogo Gos. Univ., 17 (1937), 1-26. |
[19] |
N. Maman and B. Larrouturou, Dynamical mesh adaption for two-dimensional reactive flow simulations, in "Numerical Grid Generation in Computational Fluid Dynamics and Related Fields" (Barcelona, 1991), 13-26, North-Holland, Amsterdam, 1991. |
[20] |
P. Martinez and J.-M. Roquejoffre, Convergence to critical waves in KPP equations, in preparation. |
[21] |
J.-F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal., 26 (1995), 1-20.
doi: 10.1137/S0036141093246105. |
[22] |
J. R. Norris, Long-time behaviour of heat flow : global estimates and exact asymptotics, Arch. Rational Mech. Anal., 140 (1997), 161-195.
doi: 10.1007/s002050050063. |
[23] |
J. Ortega and E. Zuazua, Large time behaviour in RN for linear parabolic equations with periodic coefficients, Asymptotic Analysis, 22 (2000), 51-85. |
[24] |
J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling waves for semi-linear parabolic equations in cylinders, Ann. IHP, Analyse non linéaire, 14 (1997), 499-552. |
[25] |
K. Uchiyama, The behaviour of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. |
[26] |
J.-L. Vazqueza and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data, Chinese Annals of Mathematics, 23, ser. B (2002), 293-310. Special issue in honor of J. L. Lions. |
show all references
References:
[1] |
B. Audoly, H. Berestycki and Y. Pomeau, Réaction-diffusion en écoulement rapide, C. R. Acad. Sci.Paris, série II, 328 (2000), 255-262. |
[2] |
M. Bages, "Équations de réaction-diffusion de type KPP: ondes pulsatoires, dynamique non triviale et applications," Ph.D thesis, Université de Toulouse, 2007. |
[3] |
M. Bages, P. Martinez and J.-M. Roquejoffre, Dynamique en grand temps pour une classe d'équations de type KPP en milieu périodique, C.R. Acad. Sci. Paris, Ser. I, 346 (2008), 1051-1056. |
[4] |
M. Bages, P. Martinez and J.-M. Roquejoffre, How travelling waves attract solutions of KPP-type equations, Trans A.M.S., to appear. |
[5] |
H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, In "Perspectives in Nonlinear Partial Differential Equations," in honor of H. Brezis, Contemp. Math., 446 Amer. Math. Soc., 101-123. |
[6] |
F. Benkhaldoun and B. Larrouturou, Numerical analysis of the two-dimensional thermodiffusive model for flame propagation, RAIRO Modèl. Math. Anal. Numér., 22 (1988), 535-560. |
[7] |
H. Berestycki and B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, vol. X, H. Brezis & J.-L. Lions eds., Pitman, London, (1991), 65-129. |
[8] |
H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré, Anal. non Linéaire, 9 (1992), 497-572. |
[9] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Memoirs of the AMS, 44 (1983), Providence, R.I. |
[10] |
P. Collet and J. P. Eckmann, Space-time behaviour in problems of hydrodynamic type: a case study, Nonlinearity, 5 (1992), 1265-1302.
doi: 10.1088/0951-7715/5/6/004. |
[11] |
U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Physica D, 146 (2000), 1-99.
doi: 10.1016/S0167-2789(00)00068-3. |
[12] |
A. Fannjiang and G. Papanicolaou, Convection Enhanced Diffusion for Periodic Flows, SIAM J. Appl. Math., 54 (1994), 333-408.
doi: 10.1137/S0036139992236785. |
[13] |
F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\RR^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[14] |
F. Hamel and L. Roques, Uniqueness and stability of monostable pulsating travelling fronts, J. European Math. Soc, to appear. |
[15] |
F. Hamel and L. Ryzhik, Non-adiabatic KPP fronts with an arbitrary Lewis number, Nonlinearity, 18 (2005), 2881-2902.
doi: 10.1088/0951-7715/18/6/024. |
[16] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840,1990, Springer. |
[17] |
A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Ann. de l'Inst. Henri Poincaré, C. Analyse non linéaire, 18 (2001), 309-358. |
[18] |
A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskowskogo Gos. Univ., 17 (1937), 1-26. |
[19] |
N. Maman and B. Larrouturou, Dynamical mesh adaption for two-dimensional reactive flow simulations, in "Numerical Grid Generation in Computational Fluid Dynamics and Related Fields" (Barcelona, 1991), 13-26, North-Holland, Amsterdam, 1991. |
[20] |
P. Martinez and J.-M. Roquejoffre, Convergence to critical waves in KPP equations, in preparation. |
[21] |
J.-F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal., 26 (1995), 1-20.
doi: 10.1137/S0036141093246105. |
[22] |
J. R. Norris, Long-time behaviour of heat flow : global estimates and exact asymptotics, Arch. Rational Mech. Anal., 140 (1997), 161-195.
doi: 10.1007/s002050050063. |
[23] |
J. Ortega and E. Zuazua, Large time behaviour in RN for linear parabolic equations with periodic coefficients, Asymptotic Analysis, 22 (2000), 51-85. |
[24] |
J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling waves for semi-linear parabolic equations in cylinders, Ann. IHP, Analyse non linéaire, 14 (1997), 499-552. |
[25] |
K. Uchiyama, The behaviour of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. |
[26] |
J.-L. Vazqueza and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data, Chinese Annals of Mathematics, 23, ser. B (2002), 293-310. Special issue in honor of J. L. Lions. |
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