Dispersive decay for the Nonlinear magnetic Schrödinger equation
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 $Σ_{s}$. By proving the resolvent estimates with weight functions and the \textquotedblleft almost equivalence\textquotedblright between $(-Δ_{A})^{\frac{s}{2}}$ and $(-Δ)^{\frac{s}{2}}$, we obtain the Strichartz estimates of $|J_{A}(t)|^{s}u:=e^{\frac{i|x|^{2}}{4t}}(-t^{2}Δ_{A})^{\frac{s}{2}}e^{\frac{-i|x|^{2}}{4t}}u$, therefore the dispersive decay estimates are obtained.
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