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February 2005 Well-posedness of the Cauchy problem for the semilinear Schrödinger equation with quadratic nonlinearity in Besov spaces
Shifu TAOKA
Hokkaido Math. J. 34(1): 65-96 (February 2005). DOI: 10.14492/hokmj/1285766209

Abstract

Well-posedness of the Cauchy problem for the semilinear Schrödinger equation with quadratic nonlinear terms is studied. By making use of Besov spaces we can improve the regularity assumption on the initial data. When the nonlinear term is $c_1u^2+c_2\bar{u}^2$, our results are as follows: When $d=1$ or $2$, for any initial data $u_0\in H^{-3/4}({\mathbb R}^d)$ there exists a unique local-in-time solution $u\in B_{2,(2,1),-|\xi|^2}^{(-3/4,1/2)}({\mathbb R}^d\times I_T)$. When $d\ge 3$, for any small data $u_0\in H^{\,\rho}({\mathbb R}^d)$, where $\rho(z)=z^{d/2-2}\log (2+z)$, there exists a unique local-in-time solution $u\in B_{2,(2,1),-|\xi|^2}^{(\,\rho,1/2)}({\mathbb R}^d\times I_T)$, and for any $u_0\in H^{s}({\mathbb R}^d)$, $s>d/2-2$, there exists a unique local-in-time solution $u\in B_{2,(2,1),-|\xi|^2}^{(s,1/2)}({\mathbb R}^d\times I_T)$. Here $I_T=(-T,T)$. We also have results for the equation with the nonlinear term $c_3u\bar{u}$.

Citation

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Shifu TAOKA. "Well-posedness of the Cauchy problem for the semilinear Schrödinger equation with quadratic nonlinearity in Besov spaces." Hokkaido Math. J. 34 (1) 65 - 96, February 2005. https://doi.org/10.14492/hokmj/1285766209

Information

Published: February 2005
First available in Project Euclid: 29 September 2010

zbMATH: 1067.35116
MathSciNet: MR2130772
Digital Object Identifier: 10.14492/hokmj/1285766209

Subjects:
Primary: 35G25
Secondary: 35Q55, 46E35

Rights: Copyright © 2005 Hokkaido University, Department of Mathematics

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Vol.34 • No. 1 • February 2005
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