标题： 非均匀非线性薛定谔方程的全局存在性和散射性 显示英文标题

标题： Global existence and scattering for the inhomogeneous nonlinear Schrödinger equation

摘要： 在本文中，我们考虑非齐次非线性薛定谔方程 $i\partial_t u +\Delta u=K(x)|u|^\alpha u,\, u(0)=u_0\in H^s({\mathbb R}^N),\, s=0,\,1,$ $N\geq 1,$ $|K(x)|+|x|^s|\nabla^sK(x)|\lesssim |x|^{-b},$ $0<b<\min(2,N-2s),$ $0<\alpha<{(4-2b)/(N-2s)}$ 。 我们获得了关于振荡初始数据的全局存在性的新结果，并在加权$L^2$-空间中得到了一个新的范围$\alpha_0(b)<\alpha<(4-2b)/N$的散射理论。 该值$\alpha_0(b)$是$N\alpha^2+(N-2+2b)\alpha-4+2b=0,$的正根，它扩展了对于$b=0$已知的 Strauss 指数。 我们的结果改进了对于$K(x)=\mu|x|^{-b}$、$\mu\in \mathbb{C}$的已知结果，并适用于更一般的势函数。 特别是，我们展示了势函数在原点和无穷远处的行为对允许的$\alpha$范围的影响。 也建立了散焦情况的一些衰减估计。 为了证明散射结果，我们给出了一个新准则，考虑了势函数$K$。 显示英文摘要

摘要： In this paper we consider the inhomogeneous nonlinear Schr\"odinger equation $i\partial_t u +\Delta u=K(x)|u|^\alpha u,\, u(0)=u_0\in H^s({\mathbb R}^N),\, s=0,\,1,$ $N\geq 1,$ $|K(x)|+|x|^s|\nabla^sK(x)|\lesssim |x|^{-b},$ $0<b<\min(2,N-2s),$ $0<\alpha<{(4-2b)/(N-2s)}$. We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted $L^2$-space for a new range $\alpha_0(b)<\alpha<(4-2b)/N$. The value $\alpha_0(b)$ is the positive root of $N\alpha^2+(N-2+2b)\alpha-4+2b=0,$ which extends the Strauss exponent known for $b=0$. Our results improve the known ones for $K(x)=\mu|x|^{-b}$, $\mu\in \mathbb{C}$ and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of $\alpha$. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential $K$.