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

### Satoshi Masaki

Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan### Jun-ichi Segata

Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan

## Abstract

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 $L^_{r}$ space where $L^_{r}={f∈S_{′}(R)∣ f _{L^_{r}}= f^ _{L_{r}}<∞}$. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to $L^_{r}$-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.

## Cite this article

Satoshi Masaki, Jun-ichi Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 2, pp. 283–326

DOI 10.1016/J.ANIHPC.2017.04.003