$u(t,x)\in\mathbb{R}^{n}$ is solution to heat equation,
$\partial_{t}u-\Delta u=0.$
Initial condition is,
$u(0,x)=u_{0}(x)\in L^{1}(\mathbb{R}^{n})$.
Rescaled function $v(t,x)$ is as follows.
$v(t,x)=t^{\frac{n}{2}}u(t,x\sqrt{t})$.
I want to find $\lim\limits_{t\rightarrow \infty}v(t,x)$.
I tried change of variable $y=x\sqrt{t}$.
Then, $u(t,y)=t^{-\frac{n}{2}}v\left(t,\frac{y}{\sqrt{t}}\right)$ and
\begin{multline} \partial_{t}u-\Delta u=-\frac{n}{2}t^{-\frac{n}{2}-1}v\left(t,\frac{y}{\sqrt{t}}\right)+t^{-\frac{n}{2}}v_{t}\left(t,\frac{y}{\sqrt{t}}\right)-\frac{1}{2}t^{-\frac{n+3}{2}}y\cdot Dv\left(t,\frac{y}{\sqrt{t}}\right)-t^{-\frac{n}{2}-1}\Delta v\left(t,\frac{y}{\sqrt{t}}\right)\\ \end{multline}
But I have no idea how to find the limit of $v$ ...