Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 68, pp. 1-5.
Title: Similarity solutions to evolution equations in one-dimensional interfaces
Authors: Mohammed Benlahsen (Univ. de Picardie Jules Verne, Amiens, France)
Ayman Eldoussouki (Univ. de Picardie Jules Verne, Amiens, France)
Mohammed Guedda (Univ. de Picardie Jules Verne, Amiens, France)
Mustapha Jazar (Univ. de Picardie Jules Verne, Amiens, France)
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.