Local existence of solutions to the cauchy problem for nonlinear schrödinger equations

Nakao Hayashi, Elena I. Kaikina

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)

Abstract

In this paper we consider the local existence to the Cauchy problem for nonlinear Schrödinger equations with power nonlinearities(*){i∂ tu+1/2Δu=N(u, ∇, u, ū, ∇ū) (t, x)∈R×R n u(O, X)=U O (X), X∈R n, where n≥2 and N=N(u, w, ū, w̄)= ∑ l0 ≥|α|+|β|+|γ|≥L1 λ αβγU α 1U α 2Π j=1 n(W j) β jΠ n k=1(W K) γKwithw=(w j) 1<j<nαβγ∈Cl o∈N,l 1, L 0 > 2. Classical energy method is useful to show local existence in time of solutions to (*) when ∂ wN is pure imaginary (see, [10, 14-16]), and in this case it is known that there exists a unique solution if U 0∈H n/2 3,0, (see [10]), whereH m's={f∈L 2;||f|| m,s=||(1+|x| 2) s/2(l -Δ) m/2f||L 2<∞}. However,if ∂ wN is not pure imaginary, there are only a few results[2,12,13] that require higher order Sobolev spaces compared with [10, 14-16] because the classical energy method does not work for the problem. Our purpose in this paper is to show local existence in time of solutions to (*) in the weighted Sobolev space H [n/2]+6,0 H [n/2]+3,2 without any size restriction on the data. Our function spaces are more natural than those used in [2,12,13].

Original languageEnglish
Pages (from-to)11-137
Number of pages127
JournalSUT Journal of Mathematics
Volume34
Issue number2
Publication statusPublished - 1998 Dec 1
Externally publishedYes

Keywords

  • Local existence
  • Local nonlinearity
  • Nonlinear schrödinger equation

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint Dive into the research topics of 'Local existence of solutions to the cauchy problem for nonlinear schrödinger equations'. Together they form a unique fingerprint.

Cite this