= Gaussian hot spot under diffusion
{c}
{title2=$u(x,t)=\left(\frac a{a+4\alpha t}\right)^{d/2}e^{-|x-x_0|^2/(a+4\alpha t)}$}
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.
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