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Preprints, Working Papers, ... Year : 2021

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 hu belongs to a classical Sobolev space, where h is a function related to this ratio. The LWP in H^s (\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 (\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.
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Dates and versions

hal-03325490 , version 1 (24-08-2021)

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Luc Molinet, Raafat Talhouk, Ibtissame Zaiter. On well-posedness for some Korteweg-De Vries type equations with variable coefficients. 2021. ⟨hal-03325490⟩
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