Local well-posedness for the gKdV equation on the background of a bounded function
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José Manuel Palacios [1]
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[1] Université de Tours; Université d’Orleans
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 1, 2023, págs. 341-396
- Idioma: inglés
- DOI: 10.4171/RMI/1345
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Resumen
We prove the local well-posedness for the generalized Korteweg–de Vries equation in Hs .R/, s > 1=2, under general assumptions on the nonlinearity f .x/, on the background of an L1t;x-function ‰.t; x/, with ‰.t; x/ satisfying some suitable conditions. As a consequence of our estimates, we also obtain the unconditional uniqueness of the solution in Hs .R/. This result not only gives us a framework to solve the gKdV equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of a periodic solution. As a direct corollary, we obtain the unconditional uniqueness of the gKdV equation in Hs .R/ for s > 1=2. We also prove global existence in the energy space H1 .R/, in the case where the nonlinearity satisfies jf 00.x/j . 1.
