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We study in this work a continuum model derived from 1Dattachment-detachment-limited (ADL) type step flow on vicinal surface, $u_t=-u^2(u^3)_{hhhh}$, where $u$, considered as a function of step height $h$,is the step slope of the surface. We formulate a notion of weak solution tothis continuum model and prove the existence of a global weak solution, whichis positive almost everywhere. We also study the long time behavior of weaksolution and prove it converges to a constant solution as time goes toinfinity. The space-time H\"older continuity of the weak solution is alsodiscussed as a byproduct.

Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime