Hokkaido Mathematical Journal

TAOKA Shifu,

Well-posedness of the Cauchy problem for the semilinearSchr\"odinger equation with quadratic nonlinearity in Besov spaces.

Hokkaido Mathematical Journal, 34 (2005) pp.65-96

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Abstract

Well-posedness of the Cauchy problem for the semilinear Schr\"odinger 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 $c_3u\bar{u}$.

MSC(Primary)35G25
MSC(Secondary)35Q55, 46E35
Uncontrolled Keywordssemilinear Schr\"odinger equation, Besov type norm, initial value problem