previous :: next

### Asymptotics of solutions to the fourth order Schrödinger type equation with a dissipative nonlinearity

Jun-ichi Segata and Akihiro Shimomura
Source: J. Math. Kyoto Univ. Volume 46, Number 2 (2006), 439-456.

#### Abstract

In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrödinger type equation with a cubic dissipative nonlinearity $\lambda |u|^{2}u$, where $\lambda$ is a complex constant satisfying $\mathrm{Im}\lambda < 0$, is studied. This nonlinearity is a long-range interaction. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as $t \to +\infty$ in a suitable function space. Our modified free dynamics decays like $(t\log t)^{-1/2}$ as $t\to \infty$.

First Page:
Primary Subjects: 35Q55
Secondary Subjects: 35C20, 35P25
Full-text: Open access