Time decay of small solutions to quadratic nonlinear Schrödinger equations in 3D

Abstract

We study large time asymptotic behavior of small solutions to the quadratic nonlinear Schrödinger equation \begin{equation} \begin{cases} i\partial _{t}u+\frac{1}{2}\Delta u=\lambda u^{2}+\mu \overline{u}^{2},\text{ }(t,x)\in {\mathbf{R}}\times {\mathbf{R}}^{3}, \\ u(0,x)=u_{0}(x),\text{ }x\in {\mathbf{R}}^{3}, \end{cases} \tag*{(0.1)} \end{equation} where $\lambda$, $\mu\in\mathbf{C}$. We prove the $\mathbf{L}^{p}(2\leq p\leq \infty )$ - time decay estimate of solutions \begin{equation*} \big\| u ( t ) \big\| _{\mathbf{L}^{p}}\leq C\big( 1+ | t | \big) ^{-\frac{3}{2}\big( 1-\frac{2}{p}\big) }\big( \left\| u_{0}\right\| _{3,0}+\left\| u_{0}\right\| _{1,2}\big) \end{equation*} under the condition that $u_{0}\in \mathbf{H}^{3,0}\cap \mathbf{H}^{1,2}$ is sufficiently small, where \begin{equation*} \mathbf{H}^{m,k}=\left\{ \phi \in \mathbf{L}^{2}:\big\| \phi \big\| _{m,k}\equiv \big\| \langle x\rangle ^{k}\langle i\nabla \rangle ^{m}\phi \big\| _{\mathbf{L}^{2}} <\infty \right\} \end{equation*} is the weighted Sobolev space and $\langle x\rangle =\sqrt{1+x^{2}}$.