Pinning and de-pinning phenomena in front propagation in heterogeneous media

Interfaces and Free Boundaries

Volumn 8, Issue 1, 2006, Pages 79-109

  Dirr, N ^a   Yip, N K ^b  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

^b PURDUE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33646424094     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/136     Document Type: Article

References (28)

1. ABEYARATNE, R. A., CHU, C., & JAMES, R. D. Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape. Philos. Mag. A 73 (1996), 457-497.
2. ALLEN, S. M., & CAHN, J. A microscopic theory for antiphase boundary motion and its application to domain coarsening. Acta Metall. 27 (1979), 1085-1095.
3. AMBROSETTI, A., & PRODI, G. A Primer of Nonlinear Analysis. Cambridge Stud. Adv. Math. 34, Cambridge Univ. Press (1993). Zbl 0818.47059 MR 1336591
4. ARAO REIS, F. D., Depinning transitions in interface growth models. Brazilian J. Phys. 33 (2003), 501-513.
5. BERTOTTI, G. Hysteresis in Magnetism. Academic Press Ser. Electromagnetism, Academic Press, San Diego (1998).
6. BHATTACHARYA, K., & CRACIUN, B. Homogenization of a Hamilton-Jacobi equation associated with the geometric motion of an interface. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 773-805. Zbl 1043.35028 MR 2006202
7. BHATTACHARYA, K., & CRACIUN, B. Effective motion of a curvature-sensitive interface through a heterogeneous medium. Interfaces Free Bound. 6 (2004), 151-173. Zbl 1061.35148 MR 2079601
8. CARPIO, A., & BONILLA, L. L. Depinning transitions in discrete reaction-diffusion equations. SIAM J. Appl. Math. 63 (2003), 1056-1082. Zbl 1035.34058 MR 1969687
9. DANCER, E. N., & HESS, P. Stability of fixed points for order-preserving discrete-time dynamical systems. J. Reine Angew. Math. 419 (1991), 125-139. Zbl 0728.58018 MR 1116922
10. FIFE, P. C., & MCLEOD, J. B. The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rat. Mech. Anal. 65 (1977), 335-361. Zbl 0361.35035 MR 0442480
11. FISHER, D. S. Collective transport in random media: from superconductors to earthquakes. Phys. Rep. 301 (1998), 113-150.
12. FRIEDMAN, A. Partial Differential Equations of Parabolic Type. Prentice-Hall (1964). Zbl 0144.34903 MR 0181836
13. GILBARG, D., & TRUDINGER, N. S. Elliptic Partial Differential Equations of Second Order. 2nd ed., Springer (1983). Zbl 0562.35001 MR 0737190
14. GRUNEWALD, N. Barkhausen effect: a stick-slip motion in a random medium. Methods Appl. Anal., to appear.
15. HENRY, D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840, Springer (1981). Zbl 0456.35001 MR 0610244
16. IOFFE, L.B., & VINOKUR, V. M. Dynamics of interfaces and dislocations in disordered media. J. Phys. C: Solid State Phys. 20 (1987), 6149-6158.
17. KARDAR, M. Nonequilibrium dynamics of interfaces and lines. Phys. Rep. 301 (1998), 85-112.
18. LIONS, P.-L., & SOUGANIDIS, P. E. Homogenization of degenerate second-order PDE in periodic and almost-periodic environments and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 667-677.
19. LUNARDI, A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser (1995). Zbl 0816.35001 MR 1329547
20. NAMAH, G., & ROQUEJOFFRE, J. M. Convergence to periodic fronts in a class of semilinear parabolic equations. Nonlinear Differential Equations Appl. 4 (1997), 521-536. Zbl 0887.35070 MR 1485736
21. NATTERMANN, T. Theory of the random field ising model. Spin Glasses and Random Fields, A. P. Young (ed.), World Sci., Singapore (1998), 277-298.
22. RUBINSTEIN, J., STERNBERG, P., & KELLER, J. B. Fast reaction, slow diffusion, and curve shortening. SIAM J. Appl. Math. 49 (1989), 116-133. Zbl 0701.35012 MR 0978829
23. RUBIO, M. A., EDWARDS, C. A., DOUGHERTY, A., & GOLLUB, J. P. Self-affine fractal interfaces from immiscible displacement in porous media. Phys. Rev. Lett. 63 (1989), 1685-1688.
24. SATTINGER, D. H. On the stability of waves of nonlinear parabolic systems. Adv. Math. 22 (1976), 312-355. Zbl 0344.35051 MR 0435602
25. SOUGANIDIS, P. E. Front propagation: theory and applications. Viscosity Solutions and Applications (Montecatini Terme, 1995), Lecture Notes in Math. 1660, Springer (1997), 186-242. Zbl 0882.35016 MR 1462703
26. XIN, J. Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity. J. Dynamical Differential Equations 3 (1991), 541-573. Zbl 0769.35033 MR 1129560
27. XIN, J. Existence and non-existence of travelling waves and reaction-diffusion front propagation in periodic media. J. Statist. Phys. 73 (1993), 893-926. MR 1251222
28. XIN, J., & ZHU, J. Quenching and propagation of bistable reaction-diffusion fronts in multidimensional periodic media. Phys. D 81 (1995), 94-110. Zbl 0900.35193 MR 1313470