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}}

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