Well-posedness for the supercritical gKdV equation

Nils Strunk

doi:10.3934/cpaa.2014.13.527

Article Contents

2014, Volume 13, Issue 2: 527-542. Doi: 10.3934/cpaa.2014.13.527

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Well-posedness for the supercritical gKdV equation

Nils Strunk^1,

1.
Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, 33501 Bielefeld

Received: September 2012

Revised: July 2013

Published: March 2014

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Abstract

In this paper we consider the supercritical generalized Korteweg-de~Vries equation $\partial_t\psi + \partial_{x x x}\psi + \partial_x(|\psi|^{p-1}\psi) = 0$, where $5 \leq p \in R$. We prove a local well-posedness result in the homogeneous Besov space $\dot B_\infty^{s_p,2}(R)$, where $s_p=\frac12-\frac{2}{p-1}$ is the scaling critical index. In particular local well-posedness in the smaller inhomogeneous Sobolev space $H^{s_p}(R)$ can be proved similarly. As a byproduct a global well-posedness result for small initial data is also obtained.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B30.

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