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

Nakao Hayashi, Elena I. Kaikina

研究成果: Article査読

12 被引用数 (Scopus)

抄録

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].

本文言語English
ページ(範囲)11-137
ページ数127
ジャーナルSUT Journal of Mathematics
34
2
出版ステータスPublished - 1998 12月 1
外部発表はい

ASJC Scopus subject areas

  • 数学 (全般)

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