Dispersive decay for the Nonlinear magnetic Schrödinger equation

Abstract

In this paper, we obtain the dispersive estimates and global well-posedness of the nonlinear magnetic Schrödinger equation in ${\mathbb{R}}^3$ with nonlinearity $|u{|}^{p-1}u$ with exponent $\frac{5}{3}<p<5$ when the initial value stays in a suitable space ${\Sigma }_s$. By proving the resolvent estimates with weight functions and the \textquotedblleft almost equivalence\textquotedblright between $(-{\Delta }_A{)}^{\frac{s}{2}}$ and $(-\Delta {)}^{\frac{s}{2}}$, we obtain the Strichartz estimates of $|J_A(t){|}^{s}u:=e^{\frac{i|x{|}^2}{4t}}(-t^2{\Delta }_A{)}^{\frac{s}{2}}{e}^{\frac{-i|x{|}^2}{4t}}u$, therefore the dispersive decay estimates are obtained.