## Fundamenal solution of heat equation

Suppose

$\displaystyle \begin{cases}u_t-\Delta u=0 \text{ in }\mathbb{R}^n\\ u(x,0)=g(x)\end{cases}$

then taking the spatial fourier transformation

$\displaystyle \hat{u}_t+4\pi^2|\xi|^2\hat{u}=0$, $\hat{u}(\xi,0)=\hat{g}(\xi)$

This means

$\displaystyle \hat{u}_t=\hat{g}(\xi)e^{-4\pi^2|\xi|^2t }$

Since $e^{-4\pi^2|\xi|^2t }=\left(\frac{1}{(4\pi t)^{n/2}}e^{-\frac{|x|^2}{4t}}\right)^\wedge$, then

$\displaystyle u=g*\frac{1}{(4\pi t)^{n/2}}e^{-\frac{|x|^2}{4t}}$