A new global Carleman estimate for Cahn–Hilliard type equation and its applications

We establish a new Carleman inequality for linear Cahn–Hilliard type equation when the norms of right hand sides are in Sobolev spaces of negative orders, and apply it to obtain the Unique Continuation Property (UCP) for Cahn–Hilliard type equation with the coefficients of terms of lower order that are not regular and null controllability for semilinear Cahn–Hilliard type equation whose semilinear term contains derivatives of first and second order of solutions.