Inverted Parabola as a Stable Steady Solution for the Surface Growth Model
ABSTRACT
For the surface growth model under molecular beam epitaxy on the real line, we show that a quadratic steady solution is stable in the sense that any small perturbation of it decays in L^\infty at the rate O(t^(-1/2)log t). The proof is based on the explicit solution operator derived from the linearized perturbation equation. This result partially justifies the merging of two neighboring islands into a bigger one. Moreover, any derivative of its solution is shown to be decaying in L^2 for sufficiently small initial data.
