= Gaussian diffusion of a smooth initial profile
{c}
{title2=$u(r,t)=(1+4t)^{-1}e^{-r^2/(1+4t)}$}
For <diffusion> with a constant <diffusion coefficient> $D>0$ and no sources, the model <flux> is $J=-D\nabla u$. A <continuity equation> gives $u_t=-\nabla\cdot J=D\Delta u$. This proportionality is a constitutive assumption about transport; the <continuity equation> expresses conservation. If $D$ varies with position, it must remain inside the <divergence>, giving $u_t=\nabla\cdot(D\nabla u)$ instead.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[Volume II, §12–4, equations 12.19–12.22: diffusion flux and local conservation]{ref}
An exact constructed <heat equation> example on the whole plane uses dimensionless coordinates with $D=1$ and the smooth initial <Gaussian function> $u(x,y,0)=e^{-r^2}$. Set $s=1+4t$. For $t\geq0$,
$$
u(x,y,t)=\frac1s e^{-r^2/s},\qquad r^2=x^2+y^2.
$$
Direct differentiation verifies both the initial condition and
$$
u_t=\Delta u=\frac{4(r^2-s)}{s^3}e^{-r^2/s}.
$$
The center decreases, while the region $r^2>s$ increases at that instant. The sign-change <circle> expands with <radius> $\sqrt{s}$. A fixed outer location can initially increase and later decrease as this <circle> passes it.
\Image[/laplacian-gaussian-diffusion-profiles.png]
{title=An exact <Gaussian function> spreads under the <heat equation> while its spatial <integral> stays fixed}
{description=Original analytic model with $D=1$. Left: radial profiles at $t=0,1/4,1$, whose peaks are $1,1/2,1/5$. Right: $u_t=\Delta u$ at $t=1/4$; the sign changes at $r=\sqrt2$. These are model predictions, not experimental measurements. The conserved quantity uses the whole-plane area element $2\pi r\,dr$, rather than the unweighted area under a radial profile.}
At time $t$, the peak is $1/s$, and the <radius> where the profile is $1/e$ times its peak is $\sqrt{s}$. The width grows as the height falls. Integration in <polar coordinates> gives
$$
\int_{\mathbb R^2}u(x,y,t)\,dx\,dy=2\pi\int_0^\infty\frac r{s}e^{-r^2/s}\,dr=\pi.
$$
Thus a decreasing peak is compatible with a conserved spatial <integral>. Dividing $u$ by $\pi$ makes a <probability density function> with coordinate <variances> $s/2=1/2+2t$, connecting the same spreading to <Brownian motion> with <diffusion coefficient> one. Equivalently the profile is evolved by the <Gaussian heat kernel> with time parameter $2t$, since that article uses generator $\tfrac12\Delta$.
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