The initial value problem for the 1-D semilinear Schrödinger equation in Besov spaces

Abstract

We define a class of Besov type spaces which is a generalization of that defined by Kenig-Ponce-Vega ([4], [5]) in their study on $KdV$ equation and nonlinear Schrödinger equation. Using these spaces, we prove the following results: the 1-dimendional semilinear Schrödinger equation with the nonlinear term $c_1{u}^2+c_2{\overline{u}}^2$ has a unique local-in-time solution for the initial data $∊B_{2,1}^{-3/4}$, and that with $cu\overline{u}$ has a unique local-in-time solution for the initial data $\in B_{2,1}^{-1/4, \sharp}$. Note that $B_{2,1}^{-1/4, \sharp}(\mathbf{R}) \supset B_{2,1}^{-1/4}(\mathbf{R}) \supset H^5(\mathbf{R})$ for any $s > -1/4$.