https://hal-cnrs.archives-ouvertes.fr/hal-03325490Molinet, LucLucMolinetIDP - Institut Denis Poisson - UO - Université d'Orléans - UT - Université de Tours - CNRS - Centre National de la Recherche ScientifiqueTalhouk, RaafatRaafatTalhoukUL - Université Libanaise, Faculté des Sciences Section (1) Hadath-BeyrouthZaiter, IbtissameIbtissameZaiterUL - Université Libanaise, Faculté des Sciences Section (1) Hadath-BeyrouthOn well-posedness for some Korteweg-De Vries type equations with variable coefficientsHAL CCSD2021KdVVariable coefficientsWell-posedness[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]Molinet, Luc2021-08-24 22:55:092022-06-26 03:12:332021-08-25 09:58:28enPreprints, Working Papers, ...https://hal-cnrs.archives-ouvertes.fr/hal-03325490/documentapplication/pdf1In 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.