September  2012, 17(6): 1639-1649. doi: 10.3934/dcdsb.2012.17.1639

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

Received  November 2011 Revised  February 2012 Published  May 2012

Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.
Citation: Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639
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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