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On well-posedness and wave operator for the gKdV equation. (English) Zbl 1305.35130
Authors’ abstract: We consider the generalized Korteweg-de Vries (gKdV) equation \(\partial_tu+\partial_x^3u + \mu\partial_x(u^{k+1})=0\), where \(k>4\) is an integer number and \(\mu=\pm1\). We give an alternative proof of the Kenig, Ponce and Vega result in [C. E. Kenig et al., Commun. Pure Appl. Math. 46, No. 4, 527–620 (1993; Zbl 0808.35128)], which asserts local and global well-posedness in \(\dot{H}^{s_k}(\mathbb{R})\), with \(s_k=(k-4)/2k\). A blow-up alternative in suitable Strichartz-type spaces is also established. The main tool is a new linear estimate. As a consequence, we also construct a wave operator in the critical space \(\dot{H}^{s_k}(\mathbb{R})\), extending the results of Côte (2006) [R. Cote, Differ. Integral Equ. 19, No. 2, 163–188 (2006; Zbl 1212.35408)].

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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