Geodesic
Similarity solutions to evolution equations in one-dimensional interfaces
Electronic journal of differential equations, Tome 2011 (2011)
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.
Classification : 70K42, 34A34, 35K55
Keywords: nonlinear dynamic, instability, similarity solution, coarsening 
@article{EJDE_2011__2011__a53,
     author = {Benlahsen,  Mohammed and Eldoussouki,  Ayman and Guedda,  Mohammed and Jazar,  Mustapha},
     title = {Similarity solutions to evolution equations in one-dimensional interfaces},
     journal = {Electronic journal of differential equations},
     year = {2011},
     volume = {2011},
     zbl = {1228.70013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a53/}
}
TY  - JOUR
AU  - Benlahsen,  Mohammed
AU  - Eldoussouki,  Ayman
AU  - Guedda,  Mohammed
AU  - Jazar,  Mustapha
TI  - Similarity solutions to evolution equations in one-dimensional interfaces
JO  - Electronic journal of differential equations
PY  - 2011
VL  - 2011
UR  - http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a53/
LA  - en
ID  - EJDE_2011__2011__a53
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%0 Journal Article
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%A Eldoussouki,  Ayman
%A Guedda,  Mohammed
%A Jazar,  Mustapha
%T Similarity solutions to evolution equations in one-dimensional interfaces
%J Electronic journal of differential equations
%D 2011
%V 2011
%U http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a53/
%G en
%F EJDE_2011__2011__a53
Benlahsen,  Mohammed; Eldoussouki,  Ayman; Guedda,  Mohammed; Jazar,  Mustapha. Similarity solutions to evolution equations in one-dimensional interfaces. Electronic journal of differential equations, Tome 2011 (2011). http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a53/