  We investigate an initial-(periodic-)boundary value problem for a continuum
equation, which is a model for motion of grain boundaries based on the
underlying microscopic mechanisms of line defects (disconnections) and
integrated the effects of a diverse range of thermodynamic driving forces. We
first prove the global-in-time existence and uniqueness of weak solution to
this initial-boundary value problem in the case with positive equilibrium
disconnection density parameter B, and then investigate the asymptotic behavior
of the solutions as B goes to zero. The main difficulties in the proof of main
theorems are due to the degeneracy of B=0, a non-local term with singularity,
and a non-smooth coefficient of the highest derivative associated with the
gradient of the unknown. The key ingredients in the proof are the energy
method, an estimate for a singular integral of the Hilbert type, and a
compactness lemma.
