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Local well-posedness for the gKdV equation on the background of a bounded function

Expositor: José Manuel Palacios - University of Tours, France
Qui 30 set 2021, 13:30 - Palestra Virtual / Online LectureSeminário de Análise e Equações Diferenciais Parciais

Resumo: In this talk we shall prove the local well-posedness for the generalized Korteweg-de Vries equation in $H^s(\Bbb R), s>1/2,$ under general assumptions on the nonlinearity $f(x)$, on the background of an $L^\infty_{t,x}$-function $\Psi(t,x)$, with $\Psi(t,x)$ satisfying some suitable conditions. As a consequence of our estimates, we also obtain the unconditional uniqueness of the solution in $H^s(\Bbb R).$ 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. We also prove global existence in the energy space $H^1(\Bbb R),$ in the case where the nonlinearity satisfies that $\vert f''(x)\vert\le 1$