  We give a detailed description in 1-D the growth of Sobolev norms for time
dependent linear generalized KdV-type equations on the circle. For most initial
data, the growth of Sobolev norms is polynomial in time for fixed analytic
potential with admissible growth. If the initial data are given in a fixed
smaller function space with more strict admissible growth conditions for
$V(x,t)$ , then the growth of previous Sobolev norms is at most logarithmic in
time.
