Hokkaido Mathematical Journal

Well-posedness of the Cauchy problem for the semilinear Schrödinger equation with quadratic nonlinearity in Besov spaces

Shifu TAOKA

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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 $c_3u\bar{u}$.

Article information

Source
Hokkaido Math. J., Volume 34, Number 1 (2005), 65-96.

Dates
First available in Project Euclid: 29 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1285766209

Digital Object Identifier
doi:10.14492/hokmj/1285766209

Mathematical Reviews number (MathSciNet)
MR2130772

Zentralblatt MATH identifier
1067.35116

Subjects
Primary: 35G25: Initial value problems for nonlinear higher-order equations
Secondary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Keywords
semilinear Schr\"odinger equation Besov type norm initial value problem

Citation

TAOKA, Shifu. Well-posedness of the Cauchy problem for the semilinear Schrödinger equation with quadratic nonlinearity in Besov spaces. Hokkaido Math. J. 34 (2005), no. 1, 65--96. doi:10.14492/hokmj/1285766209. https://projecteuclid.org/euclid.hokmj/1285766209


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