= Solution
The <heat equation> is linear and homogeneous in $u$: the operator $L=\partial_t-\lambda\partial_x^2$ satisfies $L(Au_1+Bu_2)=ALu_1+BLu_2$. Thus the <superposition principle> applies to its solutions. Spatial translations are also allowed because the coefficients are constant.
Choose $a=1/(4\lambda)$ in part (b) and multiply by $\sqrt a$ to make the initial amplitude one. Taking the two shifts $b=1$ and $b=-1$ then gives
$$
\boxed{u(x,t)=\frac1{\sqrt{1+4\lambda t}}
\left\{\exp\left[-\frac{(x+1)^2}{1+4\lambda t}\right]
+\exp\left[-\frac{(x-1)^2}{1+4\lambda t}\right]\right\}.}
$$
At $t=0$ this reproduces both initial peaks. Each term is an instance of <Gaussian initial data for the heat equation>, so their sum satisfies the equation. Its extension to negative times is a real smooth solution precisely on the connected interval
$$
\boxed{-\frac1{4\lambda}<t<0.}
$$
At the lower endpoint the peaks narrow without bound and their heights diverge; there is no finite classical continuation there. For positive times each peak spreads and decreases in height while preserving its integral.
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