## Abstract

We consider the Cauchy problem for the Schrödinger equation with a quadratic nonlinearity in one space dimensio iu_{t} + 1/2u_{xx} = t^{-a}\u_{x}|^{2}, u(0,x) = U_{0}(X), where a G (0, 1). From the heuristic point of view, solutions to this problem should have a quasilinear character when o. G (1/2, 1). We show in this paper that the solutions do not have a quasilinear character for all a G (0, 1) due to the special structure of the nonlinear term. We also prove that for a G [1/2, 1) if the initial data UQ G H ' H H ' are small, then the solution has a slow time decay such as t-^{a/2} . For o. G (0, 1/2), if we assume that the initial data U_{0} are analytic and small, then the same time decay occurs.

Original language | English |
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Pages (from-to) | XXCIX-XC |

Journal | Electronic Journal of Differential Equations |

Volume | 2001 |

Publication status | Published - 2001 Dec 1 |

Externally published | Yes |

## Keywords

- Large time behaviour
- Quadratic nonlinearity
- Schrödinger equation

## ASJC Scopus subject areas

- Analysis