A fourth-order parabolic equation modeling epitaxial thin film growth. (English) Zbl 1029.35110
Summary: We study the continuum model for epitaxial thin film growth from [T. P. Schulze, R. V. Kohn, Phys. D 132, 520-542 (1999; Zbl 0962.74522)], which is known to simulate experimentally observed dynamics very well. We show existence, uniqueness and regularity of solutions in an appropriate function space, and we characterize the existence of nontrivial equilibria in terms of the size of the underlying domain. In an investigation of asymptotical behavior, we give a weak assumption under which the \(\omega\)-limit set of the dynamical system consists only of steady states. In the one-dimensional setting we can characterize the set of steady states and determine its unique asymptotically stable element. The article closes with some illustrative numerical examples.
MSC:
| 35K35 | Initial-boundary value problems for higher-order parabolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 74K35 | Thin films |
| 76A20 | Thin fluid films |
| 35K55 | Nonlinear parabolic equations |
Keywords:
regularity of solutions; fourth-order diffusion; large time behavior; steady states; existence; uniquenessCitations:
Zbl 0962.74522References:
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