Existence of solutions for the surface electromigration equation

Abstract

We consider a model that describes electromigration in nanoconductors known as surface electromigration (SEM) equation. Our purpose here is to establish local well-posedness for the associated initial value problem in Sobolev spaces from two different points of view. In the first one, we study the pure Cauchy problem and establish local well-posedness in , s > 1/2. In the second one, we study the Cauchy problem on the background of a Korteweg–de Vries solitary traveling wave in a less regular space. To obtain our results we make use of the smoothing properties of solutions for the linear problem corresponding to the Zakharov–Kuznetsov equation for the latter problem. For the former problem we use bilinear estimates in Fourier restriction spaces introduced in [].