= Solution
For the <Gaussian initial data for the heat equation>, substitute the initial profile into the <heat-kernel solution>. Completing the square in $y$ gives
$$
a_0(y-x_0)^2+\frac{(x-y)^2}{4t}
=\left(a_0+\frac1{4t}\right)(y-\mu)^2+\frac{a_0(x-x_0)^2}{1+4a_0t},\qquad
\mu=\frac{4a_0tx_0+x}{1+4a_0t}.
$$
The remaining <Gaussian integral> yields
$$
\boxed{\theta_G(x,t)=\frac{A_0}{\sqrt{1+4a_0t}}\exp\left[-\frac{a_0(x-x_0)^2}{1+4a_0t}\right].}
$$
The centre stays at $x_0$ and the squared width grows linearly with time.
For the interval pulse, the convolution reduces to $B\int_a^bG(x-y,t)dy$. Using $u=(y-x)/(2\sqrt t)$ and the <error function> gives
$$
\boxed{\theta_I(x,t)=\frac B2\left[\operatorname{erf}\left(\frac{x-a}{2\sqrt t}\right)-\operatorname{erf}\left(\frac{x-b}{2\sqrt t}\right)\right],\qquad t>0.}
$$
This is the translated and scaled <heat equation with interval-indicator initial data>. At its jump points the $t\downarrow0$ limits are $B/2$; away from them the initial data are recovered pointwise.
For fixed $a<x<b$ at late time, both <error function> arguments are small. Since $\operatorname{erf}u=2u/\sqrt\pi+O(u^3)$,
$$
\boxed{\theta_I(x,t)=\frac{B(b-a)}{\sqrt{4\pi t}}+O(t^{-3/2})\quad(t\to\infty).}
$$
The estimate is uniform for $x\in[a,b]$. Equivalently, on this fixed interval the Gaussian in the convolution is $1+O((b-a)^2/t)$.
The spreading distance is more apparent on the growing spatial scale. For the Gaussian pulse the exponent is order one when $|x-x_0|=O(\sqrt t)$. For the interval pulse, set $x=c+\xi\sqrt t$ with $c=(a+b)/2$ and fixed $\xi$. Uniformly for bounded $\xi$, its convolution gives
$$
\theta_I(c+\xi\sqrt t,t)\sim\frac{B(b-a)}{\sqrt{4\pi t}}e^{-\xi^2/4}.
$$
Thus both nonzero pulses spread over a distance \b[$O(\sqrt t)$], while their heights are $O(t^{-1/2})$ and their total integrals remain constant. More quantitatively, the variances of their normalized profiles are $1/(2a_0)+2t$ and $(b-a)^2/12+2t$, respectively, confirming the same width law.
\Image[/past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-2-2010-ib-heat-pulses.png]
{title=Gaussian and interval initial profiles broaden under the heat equation while conserving their integrals}
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