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On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data
1. | Univ. Montpellier 2, I3M, UMR CNRS 5149, CC051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France |
2. | Graduate school of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914 |
References:
[1] |
M. Alfaro, J. Droniou and H. Matano, Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature,, J. Evol. Equ., ().
|
[2] |
M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system, J. Differential Equations, 245 (2008), 505-565.
doi: 10.1016/j.jde.2008.01.014. |
[3] |
S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallica, 27 (1979), 1084-1095. |
[4] |
G. Barles, L. Bronsard and P. E. Souganidis, Front propagation for reaction-diffusion equations of bistable type, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 9 (1992), 479-496. |
[5] |
G. Barles and F. Da Lio, A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions, Interfaces Free Bound., 5 (2003), 239-274. |
[6] |
G. Barles, H. M. Soner and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469.
doi: 10.1137/0331021. |
[7] |
G. Barles and P. E. Souganidis, A new approach to front propagation problems: Theory and applications, Arch. Rational Mech. Anal., 141 (1998), 237-296.
doi: 10.1007/s002050050077. |
[8] |
G. Bellettini and M. Paolini, Quasi-optimal error estimates for the mean curvature flow with a forcing term, Differential Integral Equations, 8 (1995), 735-752. |
[9] |
H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations," in honor of Haïm Brezis, Contemp. Math., 446, Amer. Math. Soc., Providence, RI, (2007), 101-123. |
[10] |
L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237.
doi: 10.1016/0022-0396(91)90147-2. |
[11] |
X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.
doi: 10.1016/0022-0396(92)90146-E. |
[12] |
X. Chen, Generation and propagation of interfaces for reaction-diffusion systems, Trans. Amer. Math. Soc., 334 (1992), 877-913.
doi: 10.2307/2154487. |
[13] |
X. Chen and F. Reitich, Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 164 (1992), 350-362.
doi: 10.1016/0022-247X(92)90119-X. |
[14] |
X.-Y. Chen, Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J., 21 (1991), 47-83. |
[15] |
Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geometry, 33 (1991), 749-786. |
[16] |
L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.
doi: 10.1002/cpa.3160450903. |
[17] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geometry, 33 (1991), 635-681. |
[18] |
K. Kawasaki and T. Ohta, Kinetic drumhead model of interface I, Progress of Theoretical Physics, 67 (1982), 147-163.
doi: 10.1143/PTP.67.147. |
[19] |
H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation, J. Differential Equations, 251 (2011), 3522-3557.
doi: 10.1016/j.jde.2011.08.029. |
[20] |
P. de Mottoni and M. Schatzman, Development of interfaces in $\R^n$, Proc. Roy. Soc. Edinburgh A, 116 (1990), 207-220.
doi: 10.1017/S0308210500031486. |
[21] |
P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.
doi: 10.2307/2154960. |
[22] |
H. M. Soner, Ginzburg-Landau equation and motion by mean curvature. I. Convergence, J. Geom. Anal., 7 (1997), 437-475. |
[23] |
H. M. Soner, Ginzburg-Landau equation and motion by mean curvature. II. Development of the initial interface, J. Geom. Anal., 7 (1997), 477-491. |
show all references
References:
[1] |
M. Alfaro, J. Droniou and H. Matano, Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature,, J. Evol. Equ., ().
|
[2] |
M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system, J. Differential Equations, 245 (2008), 505-565.
doi: 10.1016/j.jde.2008.01.014. |
[3] |
S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallica, 27 (1979), 1084-1095. |
[4] |
G. Barles, L. Bronsard and P. E. Souganidis, Front propagation for reaction-diffusion equations of bistable type, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 9 (1992), 479-496. |
[5] |
G. Barles and F. Da Lio, A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions, Interfaces Free Bound., 5 (2003), 239-274. |
[6] |
G. Barles, H. M. Soner and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469.
doi: 10.1137/0331021. |
[7] |
G. Barles and P. E. Souganidis, A new approach to front propagation problems: Theory and applications, Arch. Rational Mech. Anal., 141 (1998), 237-296.
doi: 10.1007/s002050050077. |
[8] |
G. Bellettini and M. Paolini, Quasi-optimal error estimates for the mean curvature flow with a forcing term, Differential Integral Equations, 8 (1995), 735-752. |
[9] |
H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations," in honor of Haïm Brezis, Contemp. Math., 446, Amer. Math. Soc., Providence, RI, (2007), 101-123. |
[10] |
L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237.
doi: 10.1016/0022-0396(91)90147-2. |
[11] |
X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.
doi: 10.1016/0022-0396(92)90146-E. |
[12] |
X. Chen, Generation and propagation of interfaces for reaction-diffusion systems, Trans. Amer. Math. Soc., 334 (1992), 877-913.
doi: 10.2307/2154487. |
[13] |
X. Chen and F. Reitich, Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 164 (1992), 350-362.
doi: 10.1016/0022-247X(92)90119-X. |
[14] |
X.-Y. Chen, Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J., 21 (1991), 47-83. |
[15] |
Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geometry, 33 (1991), 749-786. |
[16] |
L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.
doi: 10.1002/cpa.3160450903. |
[17] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geometry, 33 (1991), 635-681. |
[18] |
K. Kawasaki and T. Ohta, Kinetic drumhead model of interface I, Progress of Theoretical Physics, 67 (1982), 147-163.
doi: 10.1143/PTP.67.147. |
[19] |
H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation, J. Differential Equations, 251 (2011), 3522-3557.
doi: 10.1016/j.jde.2011.08.029. |
[20] |
P. de Mottoni and M. Schatzman, Development of interfaces in $\R^n$, Proc. Roy. Soc. Edinburgh A, 116 (1990), 207-220.
doi: 10.1017/S0308210500031486. |
[21] |
P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.
doi: 10.2307/2154960. |
[22] |
H. M. Soner, Ginzburg-Landau equation and motion by mean curvature. I. Convergence, J. Geom. Anal., 7 (1997), 437-475. |
[23] |
H. M. Soner, Ginzburg-Landau equation and motion by mean curvature. II. Development of the initial interface, J. Geom. Anal., 7 (1997), 477-491. |
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