A fourth-order parabolic equation modeling expitaxial thin film growth

Journal of Mathematical Analysis and Applications

Volumn 286, Issue 2, 2003, Pages 459-490

  King, Belinda B ^a   Stein, Oliver ^b   Winkler, Michael ^b  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

^b RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

Fourth order diffusion; Large time behavior; Steady states; Thin film growth

Indexed keywords

EID: 0242677020     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-247X(03)00474-8     Document Type: Article

References (34)

1. E. Beretta, M. Bertsch, R. Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal. 129 (1995) 175-200.
2. F. Bernis, Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equations 1 (1996) 337-368.
3. F. Bernis, A. Friedman, Higher order nonlinear parabolic equations, J. Differential Equations 83 (1990) 179-206.
4. J.A. Burns, B.B. King, O. Stein, Computational methods for identification and control of nanoscale materials, ICAM Report 00-03-01, Virginia Tech, 2000.
5. E.A. Carlen, M.C. Carvalho, E. Orlandi, A simple proof of stability of fronts for the Cahn-Hilliard equation, Comm. Math. Phys. 224 (2001) 323-340.
6. J. Carr, M.E. Gurtin, M. Slemrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal. 86 (1984) 317-351.
7. S. Das Sarma, S.V. Ghaisas, Solid-on-solid rules and models for nonequilibrium growth in 2 + 1 dimensions, Phys. Rev. Lett. 69 (1992) 3762-3765.
8. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
9. S.F. Edwards, D.R. Wilkinson, The surface statistics of a granular aggregate, Proc. Roy. Soc. London Ser. A 381 (1982) 17-31.
10. C.M. Elliot, S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96 (1986) 339-357.
11. A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, New York, 1969.
12. M. Grinfeld, A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 351-370.
13. M. Grinfeld, A. Novick-Cohen, The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor, Trans. Amer. Math. Soc. 351 (1999) 2375-2406.
14. C. Grossmann, H.-G. Roos, Numerik Partieller Differentialgleichungen, Teubner, Stuttgart, 1992.
15. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981.
16. C. Herring, Surface tension as a motivation for sintering, in: W.E. Kingston (Ed.), The Physics of Powder Metallurgy, McGraw-Hill, New York, 1951.
17. H. Kielhöfer, Pattern formation of the stationary Cahn-Hilliard model, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997) 1219-1243.
18. A. Lunardi, E. Sinestrari, W. von Wahl, A semigroup approach to the time dependent parabolic initial-boundary value problem, J. Differential Equations 5 (1992) 1275-1306.
19. S. Maier-Paape, U. Miller, Path-following the equilibria of the Cahn-Hilliard equation on the square, Comput. Visual. Sci. 5 (2002) 115-138.
20. K. Mischaikow, Global asymptotic dynamics of gradient-like bistable equations, SIAM J. Math. Anal. 26 (1995) 1199-1224.
21. W.W. Mullins, Theory of thermal grooving, J. Appl. Phys. 28 (1957) 333-339.
22. S.A. Nazarov, B.A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Gruyter, Berlin, 1994.
23. A. Novick-Cohen, On Cahn-Hilliard type equations, Nonlinear Anal. 15 (1990) 797-814.
24. A. Novick-Cohen, L.A. Peletier, Steady states of the one-dimensional Cahn-Hilliard equation, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 1071-1098.
25. M. Ortiz, E.A. Repetto, H. Si, A continuum model of kinetic roughening and coarsening in thin films, J. Mech. Phys. Solids 47 (1999) 697-730.
26. T.P. Schulze, R.V. Kohn, A geometric model for coarsening during spiral-mode growth of thin films, Phys. D 132 (1999) 520-542.
27. O. Stein, On epitaxial thin film growth and control of nanoscale materials, ICAM Report 00-04-01, Virginia Tech, 2000.
28. O. Stein, The role of the transition function in a continuum model for kinetic roughening and coarsening in thin films, ICAM Report 99-07-02, Virginia Tech, 1999.
29. A.E. Taylor, Introduction to Functional Analysis, Wiley, 1958.
30. R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, in: Studies in Mathematics and Its Applications, Vol. 2, North-Holland, Amsterdam, 1977.
31. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Barth, Heidelberg, 1995.
32. M. Winkler, Some results on degenerate parabolic equations not in divergence form, Ph.D. Thesis, Aachen, 2000.
33. M. Winkler, Remarks on a finite element approximation for a fourth order nonlinear parabolic equation, Preprint, Aachen, 2000.
34. A. Zangwill, Some causes and a consequence of epitaxial roughening, J. Crystal Growth 163 (1996) 8-21.