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A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach

Supported in part by JSPS, Grant-in-Aid for Scientific Research (C) #25400176

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  • We study the nonlinear Schrödinger equation (NLS)

    $\partial_t u +i \Delta u = i\lambda |u|^{p-1} u$

    in $\mathit{\boldsymbol{R}}^{1+n}$, where $n\ge 3$, $p>1$, and $\lambda \in \mathit{\boldsymbol{C}}$. We prove that (NLS) is locally well-posed in $H^s$ if $1<s<\min\{4;n/2\}$ and $\max\{1;s/2\}< p< 1+4/(n-2s)$. To obtain a good lower bound for $p$, we use fractional order Besov spaces for the time variable. The use of such spaces together with time cut-off makes it difficult to derive positive powers of time length from nonlinear estimates, so that it is difficult to apply the contraction mapping principle. For the proof we improve Pecher's inequality (1997), which is a modification of the Strichartz estimate, and apply this inequality to the nonlinear problem together with paraproduct formula.

    Mathematics Subject Classification: 35Q55, 35Q41.


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