Unique continuation and approximate controllability for a degenerate   parabolic equation
  This paper studies unique continuation for weakly degenerate parabolic equations in one space dimension. A new Carleman estimate of local type is obtained to deduce that all solutions that vanish on the degeneracy set, together with their conormal derivative, are identically equal to zero. An approximate controllability result for weakly degenerate parabolic equations under Dirichlet boundary condition is deduced. 
Piermarco Cannarsa and Jacques Tort and Masahiro Yamamoto
math.AP cs.SY math.OC