Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation

Masaki, Satoshi; Segata, Jun-ichi

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 2, pp. 283-326.

In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg–de Vries (gKdV) equation in the scale critical ${\hat{L}}^r$ space where ${\hat{L}}^r=\{f\in {\mathcal{S}}^{\prime }(\mathbb{R})|{\Vert f\Vert }_{{\hat{L}}^r}={\Vert \hat{f}\Vert }_{L^{r^{\prime }}}<\infty \}$. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to ${\hat{L}}^r$-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.

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