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

On the Growth in time of Sobolev Norms for Time Dependent Linear Generalized KdV-type Equations