Gaussian hot spot under diffusion ($u(x,t)={\left(\frac{a}{a+4\alpha t}\right)}^{d/2}{e}^{-|x-x_0{|}^2/(a+4\alpha t)}$)

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.