the backstepping approach we recover the null controllability for the
heat equations with variable coefficients in space in one dimension and
prove that these equations can be stabilized in finite time by means of
periodic time-varying feedback laws. To this end, on one hand, we
provide a new proof of the well-posedness and the ``optimal"
bound with respect to damping constants for the solutions of the kernel
equations; this allows one to deal with variable coefficients, even
with a weak regularity of these coefficients. On another hand, we
establish the well-posedness and estimates for the heat equations
with a nonlocal boundary condition at one side.