Regularity in time of solutions to nonlinear schrödinger equations

Nakao Hayashi, Keiichi Kato

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

In this paper we consider the regularity of solutions to nonlinear Schrödinger equations (NLS), i ∂tu + 1 2 Δu = F(u, u), (t, x) ∈ R × Rn, u(0) = φ, x ∈ Rn, where F is a polynomial of degree p with complex coefficients. We prove that if the initial function φ is in some Gevrey class, then there exists a positive constant T such that the solution u of NLS is in the Gevrey class of the same order as in the initial data in time variable t ∈[-T, T]0. In particilar, we show that if the initial function φ has an analytic continuation on the complex domain Γ A1, A2 = (z ∈ Cn; zj=xj+iyj, -∞ < xj < + ∞, -A2-(tan α) |xj| <yj < A2 + (tan α) |xj| j = 1, 2., n, A2 > 0) (see Fig. 1), where 0 < α = sin-1A1 < π/2 and 0 < A1 < 1, then there exists positive constants T and β such that the solution u of NLS is analytic in time variable t ∈ [-T, T]0 and has an analytic continuation on (z0 = t + iτ; |arg z0| < β <π/2, |t|<T) where sin β < Min (√2A1/(1 + √2A1), 2A2/(3A2 + [formula](1 + R))) when |x| < R.

Original languageEnglish
Pages (from-to)253-277
Number of pages25
JournalJournal of Functional Analysis
Volume128
Issue number2
DOIs
Publication statusPublished - 1995 Mar
Externally publishedYes

ASJC Scopus subject areas

  • Analysis

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