On well-posedness for some Korteweg-De Vries type equations with variable coefficients

Abstract

In this paper, KdV-type equations with time- and space-dependent coefficients are considered. Assuming that the dispersion coefficient in front of $u_{xxx}$ is positive and uniformly bounded away from the origin and that a primitive function of the ratio between the anti-dissipation and the dispersion coefficients is bounded from below, we prove the existence and uniqueness of a solution $u$ such that $h u$ belongs to a classical Sobolev space, where $h$ is a function related to this ratio. The LWP in $H^s(\mathbb{R})$, $s>1/2$, in the classical (Hadamard) sense is also proven under an assumption on the integrability of this ratio. Our approach combines a change of unknown with dispersive estimates. Note that previous results were restricted to $H^s(\mathbb{R})$, $s>3/2$, and only used the dispersion to compensate the anti-dissipation and not to lower the Sobolev index required for well-posedness.