 Gaussian hot spot under diffusion

On ${\mathbb{R}}^d$, the heat equation with initial Gaussian function $u_0(x)=e^{-|x-x_0{|}^2/a}$, $a>0$, has the displayed solution. Convolving the initial profile with the heat kernel adds the Gaussian width parameters; conservation of its spatial integral fixes the prefactor. Direct differentiation provides another proof. The width increases while the peak decreases, with no loss of the total integral.