Electron. J. Diff. Equ., Vol. 2011 (2011), No. 68, pp. 1-5.

Similarity solutions to evolution equations in one-dimensional interfaces

Mohammed Benlahsen, Ayman Eldoussouki, Mohammed Guedda, Mustapha Jazar

Abstract:
In this note, we study the evolution equation
$$ \partial_t h = -\nu\partial^2_x h-K\partial^4_x h +\lambda_1(\partial_x h)^2-\lambda_2\partial^2_x(\partial_x h)^2. $$
which was introduced by Munoz-Garcia [9] in the context of erosion by ion beam sputtering. We obtain an analytic solution that has the similarity form, which is used in obtaining the coarsening behavior. This solution has amplitude and wavelength that increase like $\ln(t)$ and $ \sqrt{t\ln(t)}$, respectively.

Submitted April 15, 2011. Published May 20, 2011.
Math Subject Classifications: 70K42, 34A34, 35K55.
Key Words: Nonlinear dynamic; instability; similarity solution; coarsening.

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Mohammed Benlahsen
LPMC, Department of Physic, Université de Picardie Jules Verne
33, rue saint-Leu, Amiens, France
email: mohammed.benlahsen@u-picardie.fr
Ayman Eldoussouki
LAMFA, CNRS UMR 6140, Department of Mathematics
Université de Picardie Jules Verne
33, rue saint-Leu, Amiens, France
email: ayman.eldoussouki@u-picardie.fr
Mohammed Guedda
LAMFA, CNRS UMR 6140, Department of Mathematics
Université de Picardie Jules Verne
33, rue saint-Leu, Amiens, France
email: guedda@u-picardie.fr
Mustapha Jazar
LaMA-Liban, Lebanese University
P.O. Box 37 Tripoli via Beirut, Lebanon
email: mjazar@laser-lb.org

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