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

TL;DR

This paper analyzes how Sobolev norms grow over time for solutions to time-dependent linear generalized KdV equations on the circle, showing polynomial growth for typical data and logarithmic growth under stricter conditions.

Contribution

It provides a detailed description of Sobolev norm growth for 1-D linear generalized KdV equations with time-dependent potentials, including new bounds under various initial data and potential conditions.

Findings

Sobolev norms grow polynomially for most initial data with fixed analytic potential.

Under stricter conditions on initial data and potential, Sobolev norms grow at most logarithmically.

The results clarify the long-term behavior of solutions to time-dependent linear KdV-type equations.

Abstract

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.