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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

Matthieu Alfaro 1, and  Hiroshi Matano 2,

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.
Keywords: Singular perturbation, reaction-diffusion system., asymptotic expansion, front profile.
Mathematics Subject Classification: Primary: 35B25, 35C20, 35R3.
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 & 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., ().   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

X. Chen, Generation and propagation of interfaces for reaction-diffusion systems, Trans. Amer. Math. Soc., 334 (1992), 877-913. doi: 10.2307/2154487.  Google Scholar

[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.  Google Scholar

[14]

X.-Y. Chen, Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J., 21 (1991), 47-83.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geometry, 33 (1991), 635-681.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589. doi: 10.2307/2154960.  Google Scholar

[22]

H. M. Soner, Ginzburg-Landau equation and motion by mean curvature. I. Convergence, J. Geom. Anal., 7 (1997), 437-475.  Google Scholar

[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.  Google Scholar

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., ().   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

X. Chen, Generation and propagation of interfaces for reaction-diffusion systems, Trans. Amer. Math. Soc., 334 (1992), 877-913. doi: 10.2307/2154487.  Google Scholar

[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.  Google Scholar

[14]

X.-Y. Chen, Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J., 21 (1991), 47-83.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geometry, 33 (1991), 635-681.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589. doi: 10.2307/2154960.  Google Scholar

[22]

H. M. Soner, Ginzburg-Landau equation and motion by mean curvature. I. Convergence, J. Geom. Anal., 7 (1997), 437-475.  Google Scholar

[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.  Google Scholar

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