Elementary computation about NLS

Question

I was trying to show $$\lim_{t \rightarrow \infty} \int_{|x|<kt} |u(t,x)|^2 dx = \int_{|\xi|<k/2} |\hat{u_{0}}(\xi)|^2 d \xi$$ where $$ u(t,x)=e^{it\Delta} u_{0}(x), \quad \textrm{in} \quad \mathbb{R}_{+} \times \mathbb{R}^3$$ I just tried to compute the integral in the left hand side explicitly, then I got $$ \int_{|x|<kt} t^{-3} \left| \int e^{i\frac{|y|^2}{4t}} e^{-i \frac{x}{2t} \cdot y } u_{0}(y) dy \right|^2 dx ,$$ so I could get the result if $e^{i|y|^2/(4t)}$ is just replaced with $1$. But I have no idea about how to proceed from here. Any help will be appreciated. Thank you.