![enter image description here](https://i.stack.imgur.com/SPEYr.jpg)

Here $u\_0(\xi)$:=$u(\xi,0)$ where $x(0)=\xi$

It is clear from the picture that the characteristics go inwards towards $x=0$ as $t \rightarrow \infty$. I also understand that $u$ is constant on these curves. However, I do not understand why we can infer from this that the solution gets 'focussed' ( as displayed in the $u$-$x$ graph) around $x=0$ as $t \rightarrow \infty$.

$\frac{\partial u}{\partial t}-xt\frac{\partial u}{\partial x}=0$; $x \in \mathbb{R}$, $t>0$

$u(x,0)=u\_0(x)$; $x\in \mathbb{R}$ 

$-te^-\frac{t^2}{2}\sqrt \pi$