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

Junichi Segata, Akihiro Shimomura

Research output: Contribution to journalArticle

6 Citations (Scopus)

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 λ|u|2u, where λ is a complex constant satisfying Im λ < 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 → +∞ in a suitable function space. Our modified free dynamics decays like (t log t)-1/2 as t → ∞.

Original languageEnglish
Pages (from-to)439-456
Number of pages18
JournalKyoto Journal of Mathematics
Volume46
Issue number2
DOIs
Publication statusPublished - Jan 1 2006
Externally publishedYes

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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