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 Schr\"odinger 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.