Global theory for NLS in the weighted spaces II: final data problem

In this paper, we study the final data problem for the defocusing nonlinear Schrödinger equation (NLS) $$i{\mathrm{\partial }}_{t}u+\frac{1}{2}\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{p}u$$ in weighted spaces ${\dot{\mathrm{\Sigma }}}^s({\mathbb{R}}^d):=L_x^2({\mathbb{R}}^d;\mathrm{\mid }x{\mathrm{\mid }}^{2s}\mathrm{d}x)$. Let $s_c=\frac{d}{2}-\frac{2}{p}$. Under the scaling of the NLS, ${\dot{\mathrm{\Sigma }}}^{-s_c}({\mathbb{R}}^d)$ is the scaling-critical weighted space. In three dimensions, within the range of exponents considered in this paper, we show that the final data problem is locally well-posed for $s\ge -s_c$ and ill-posed for $s<-s_c$. This is opposite to the corresponding initial data problem, which is locally well-posed for $s\le -s_c$ and ill-posed for $s>-s_c$. For the three-dimensional quadratic equation, we further obtain a unique global solution for every radial final datum in the critical space ${\dot{\mathrm{\Sigma }}}^{\frac{1}{2}}({\mathbb{R}}^3)$.