Analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations

Nakao Hayashi, Keiichi Kato

Research output: Contribution to journalArticle

31 Citations (Scopus)

Abstract

In this paper we consider analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations {i∂tu + 1/2△u = λ|u|2pu, (t, x) ∈ ℝ × ℝn, u(0, x) = φ, x ∈ ℝn, (1) where λ ∈ ℂ, p ∈ ℕ. We prove that if φ satisfies ∥e|x|2φ∥ Η[n/2]+1 < ∞, (2) then there exists a unique solution u(t, x) of (1) and positive constants Τ, C0, C1 such that u(t, x) is analytic in time and space variables for t ∈ [-Τ,Τ] \ {0} and x ∈ Ω = {x; |x| < R} and has an analytic continuation U(z0, z) on {z0 = t + iτ; -C0t2 < τ < C0t2,t ∈ [-Τ,Τ] \ {0}} and {z = x + iy; -C1|t| < y < C1(t),(t, x) ∈ [-Τ,Τ] \ {0} × Ω} . In the case n = 1,2,3 the condition (2) can be relaxed as follows: ∥e|x|2φ∥ Ηm < ∞, where m = 0 if n = 1, p = 1, m = 1 if n = 2. p ∈ ℕ and m = 1 if n = 3, p = 1.

Original languageEnglish
Pages (from-to)273-300
Number of pages28
JournalCommunications in Mathematical Physics
Volume184
Issue number2
DOIs
Publication statusPublished - 1997 Jan 1
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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