Generation and propagation of interfaces for reaction-diffusion equations. (English) Zbl 0765.35024

The author considers the reaction-diffusion equation \[ \partial_ t u - \Delta u + (1/\varepsilon^ 2)\Phi (u)=0 \] on \({\mathbf R}^ N \times {\mathbf R}^ +\), where \(\Phi\) is the derivative of a bistable potential. He shows for initial conditions with values in both domains of attraction of the potential that an interface will develop in time \(O(\varepsilon ^ 2 | \text{ln} \varepsilon | )\). Depending on whether the wells of the potential have equal depth or not, he obtains a different propagation behavior of this interface (normal velocity equal to its mean curvature in the first case, constant speed proportional to the difference of the depths of the wells in the latter case). Also, the disappearance and nonextinction of the interface are treated.
Reviewer: G.Hetzer (Auburn)


35K57 Reaction-diffusion equations
Full Text: DOI


[1] Allen, S.; Cahn, J., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27, 1084-1095 (1979)
[2] Aronson, D. G.; Weinberger, H. F., Nonlinear diffusion in population genetics, combustion, and nerve propagation, (Goldstein, J. A., Partial Differential Equation and Related Topics. Partial Differential Equation and Related Topics, Lecture Notes in Mathematics, Vol. 446 (1975), Springer-Verlag: Springer-Verlag New York), 5-49 · Zbl 0325.35050
[3] Aronson, D. G.; Weinberger, H. F., Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30, 33-76 (1978) · Zbl 0407.92014
[4] Brakke, K. A., The Motion of a Surface by its Mean Curvature (1978), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0386.53047
[5] Bronsard, L.; Kohn, R. V., Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90, 211-237 (1991) · Zbl 0735.35072
[6] Bronsard, L.; Khon, R. V., On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43, 983-997 (1990) · Zbl 0761.35044
[7] Carr, J.; Pego, R., Very slow phase separation in one dimension, (Rascle, M., Lecture Notes in Physics, Vol. 344 (1989)), 216-266 · Zbl 0991.35515
[9] Chen, T. G.; Giga, Y.; Goto, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom., 33, 749-786 (1991) · Zbl 0696.35087
[10] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0042.32602
[12] DeMottoni, P.; Schatzman, M., Evolution géométrique d’interfaces, C. R. Acad. Sci. Sér. I Math., 309, 453-458 (1989) · Zbl 0698.35078
[13] Evans, L. C.; Spruck, J., Motion of level set by mean curvature I, J. Diff. Geom., 33, 635-681 (1991) · Zbl 0726.53029
[15] Fife, P. C., Diffusive waves in nonhomogeneous media, (Proc. Edinburg Math. Soc., 32 (1989)), 291-315 · Zbl 0701.35085
[16] Fife, P. C., Dynamics of internal layers and diffusive interfaces, (CCMS-NSF Regional Conf. Ser. in Appl. Math. (1988), SIAM: SIAM Philadelphia) · Zbl 0684.35001
[17] Fife, P. C., Long time behavior of solutions of bistable diffusion equations, Arch. Rational Mech. Anal., 70, 31-46 (1979) · Zbl 0435.35045
[18] Fife, P. C.; Hsiao, L., The generation and propagation of internal layers, Nonlinear Anal., 12, 19-41 (1988) · Zbl 0685.35055
[19] Fife, P. C.; McLeod, B., The approach of solutions of nonlinear diffusion equation to travelling front solutions, Arch. Rational Mech. Anal., 65, 335-361 (1977) · Zbl 0361.35035
[20] Fusco, G., A geometric approach to the dynamics of \(ut = ε^2uxx + ƒ(u)\) for small ε, (Kirchgassner, Lecture Notes in Physics, Vol. 359 (1990), Springer-Verlag), 53-73
[21] Fusco, G.; Hale, J. K., Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynamics Differential Equations, 1, No. 1, 75-94 (1989) · Zbl 0684.34055
[22] Gage, M.; Hamilton, R., The shrinking of convex plane curves by the heat equation, J. Differential Geom., 23, 69-96 (1986) · Zbl 0621.53001
[23] Grayson, M., The heat equation shrinks embedded plane curves to points, J. Diff. Geom., 26, 285-314 (1987) · Zbl 0667.53001
[24] Hamilton, R. S., Three Manifolds with positive Ricci Curvature, J. Diff. Geom., 17, 255-306 (1982) · Zbl 0504.53034
[25] Huisken, G., Flow by mean curvature of convex surface into surface, J. Diff. Geom., 20, 237-266 (1984) · Zbl 0556.53001
[26] Rubinstein, J.; Sternberg, P.; Keller, J. B., Fast reaction, slow diffusion and curve shorting, SIAM J. Appl. Math., 49, 116-133 (1989) · Zbl 0701.35012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.