Weak Solution of a Continuum Model For Vicinal Surface in The Attachment-Detachment-Limited Regime

We study in this work a continuum model derived from 1D attachment-detachmentlimited (ADL) type step flow on vicinal surface, ut = −u 2(u3)hhhh, where u, considered as a function of step height h, is the step slope of the surface. We formulate a notion of weak solution to this continuum model and prove the existence of a global weak solution, which is positive almost everywhere. We also study the long time behavior of weak solution and prove it converges to a constant solution as time goes to infinity. The space-time Hölder continuity of the weak solution is also discussed as a byproduct.