Generation and propagation of interfaces in reaction-diffusion systems

Author:
Xinfu Chen

Journal:
Trans. Amer. Math. Soc. **334** (1992), 877-913

MSC:
Primary 35R35; Secondary 35K57

DOI:
https://doi.org/10.1090/S0002-9947-1992-1144013-3

MathSciNet review:
1144013

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the asymptotic behavior, as , of the solution of the second initial-boundary value problem of the reaction-diffusion system:

**[1]**S. Allen and J. Cahn,*A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsing*, Acta Metall.**27**(1979), 1084-1095.**[2]**D. G. Aronson and H. F. Weinberger,*Nonlinear diffusion in population genetics, combustion, and nerve propagation*, Partial Differential Equations and Related Topics (J. A. Goldstein, ed.), Lecture Notes in Math., vol. 446, Springer-Verlag, New York, 1975, pp. 5-49. MR**0427837 (55:867)****[3]**-,*Multidimensional nonlinear diffusion arising in population genetics*, Adv. Math.**30**(1978), 33-76. MR**511740 (80a:35013)****[4]**G. Barles,*Remarks on a flame propagation model*, Rapport INRIA, #464, 1985.**[5]**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. MR**1101239 (92d:35037)****[6]**-,*On the slowness of phase boundary motion in one space dimension*, Comm. Pure. Appl. Math.**43**(1990), 983-987. MR**1075075 (91f:35023)****[7]**J. Carr and R. Pego,*Very slow phase separation in one dimension*, (M. Rascle, ed.), Lecture Notes in Phys., vol. 344, Springer-Verlag, 1989, pp. 216-226. MR**1036071 (91e:35120)****[8]**-,*Invariant manifolds for metastable pattern in*, Proc. Roy. Soc. Edinburgh**116**(1990), 133-160. MR**1076358 (91i:35009)****[9]**Xinfu Chen,*Generation and propagation of interface in reaction-diffusion equations*, J. Differential Equations**96**(1992), 116-141. MR**1153311 (92m:35129)****[10]**Xu-Yan Chen,*Dynamics of interfaces in reaction diffusion systems*, Hiroshima Math. J.**21**(1991), 47-83. MR**1091432 (91m:35125)****[11]**Y. G. Chen, Y. Giga, and S. Goto,*Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations*, J. Differential Geom.**33**(1991), 749-786. MR**1100211 (93a:35093)****[12]**R. Courant and D. Hilbert,*Methods of mathematical physics*, Vol. II, Wiley, New York, 1962. MR**1013360 (90k:35001)****[13]**M. G. Crandall, L. C. Evans, and P. L. Lions,*Some properties of viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc.**282**(1984), 487-502. MR**732102 (86a:35031)****[14]**P. DeMottoni and M. Schatzman,*Evolution géométrique d'interfaces*, C. R. Acad. Sci. Paris Sér. I Math.**309**(1989), 453-458. MR**1055457 (91f:35137)****[15]**-,*Development of interfaces in*, preprint.**[16]**L. C. Evans, H. M. Soner, and P. E. Souganidis,*The Allen-Cahn equation and the generalized motion by mean curvature*, preprint.**[17]**L. C. Evans and J. Spruck,*Motion of level set by mean curvature*. I, J. Differential Geom.**33**(1991), 635-681. MR**1100206 (92h:35097)****[18]**P. C. Fife,*Dynamics of internal layers and diffusive interfaces*, CBMS-NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, Pa., 1988. MR**981594 (90c:80012)****[19]**P. C. Fife and L. Hsiao,*The generation and propagation of internal layers*, Nonlinear Anal.**70**(1988), 31-46. MR**924750 (89c:35078)****[20]**P. C. Fife and B. McLeod,*The approach of solutions of nonlinear diffusion equation to travelling front solutions*, Arch. Rational Mech. Anal.**65**(1977), 335-361. MR**0442480 (56:862)****[21]**P. C. Fife and J. Tyson,*Target patterns in a realistic model of the Belousov-Zhabotinskii reaction*, J. Chem. Phys.**73**(1980), 2224-2237. MR**583644 (82a:80004)****[22]**A. Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Englewood Cliffs, N.J., 1964. MR**0181836 (31:6062)****[23]**G. Fusco,*A geometric approach to the dynamics of**for small*(K. Kirchgassner, ed.), Lecture Notes in Phys., vol. 359, Springer-Verlag, 1990, pp. 53-73. MR**1062209 (91m:35018)****[24]**G. Fusco and J. K. Hale,*Slow-motion manifolds, dormant instability, and singular perturbations*, J. Dynamics Differential Equations**1**(1989), 75-94. MR**1010961 (90i:35131)****[25]**Y. Giga, S. Goto, and H. Ishii,*Global existence of weak solutions for interface equations coupled with diffusion equations, IMA Preprint*#806, University of Minnesota, 1991.**[26]**D. Hilhorst, Y. Nishiura, and M. Mimura,*A free boundary problem arising from reaction-diffusion system*, preprint, 1990.**[27]**O. A. Ladyzhenskaja, V. A. Solonnikov, and N. N. Ural'ceva,*Linear and quasilinear equations of parabolic type*, Amer. Math. Soc., Providence, R.I., 1968.**[28]**T. Ohta, M. Mimura, and R. Kobayashi,*Higher dimensional localized patterns in excitable media*, Physics D**34**(1989), 115-144. MR**982383 (90c:35121)****[29]**S. Osher and J. A. Sethian,*Fronts propagating with curvature dependent speed*:*algorithms based on Hamilton-Jacobi formulations*, J. Comput. Phys.**79**(1988), 12-49. MR**965860 (89h:80012)****[30]**J. Rubinstein, P. Sternberg, and J. B. Keller,*Fast reaction, slow diffusion and curve shorting*, SIAM J. Appl. Math.**49**(1989), 116-133. MR**978829 (89m:35117)****[31]**J. Smoller,*Shock waves and reaction-diffusion equations*, Springer-Verlag, New York, 1982. MR**1301779 (95g:35002)****[32]**J. A. Sethian,*Curvature and evolution of fronts*, Comm. Math. Phys.**101**(1985), 487-499. MR**815197 (87d:58032)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35R35,
35K57

Retrieve articles in all journals with MSC: 35R35, 35K57

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1144013-3

Keywords:
Reaction-diffusion systems,
generation of interface,
propagation of interface

Article copyright:
© Copyright 1992
American Mathematical Society