  We study in this work a continuum model derived from 1D
attachment-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 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 H\"older continuity of the weak solution is also
discussed as a byproduct.
