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

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 $c_3u\bar{u}$.