Solution
= Solution
For $a=0$ and $\kappa>0$, the <heat kernel> gives
$$
\boxed{u(t,x)=\int_{-\infty}^{\infty}K_t(x-y)u_0(y)\,dy}.
$$
As $t\downarrow0$, $K_t$ is a Gaussian of total mass one whose width is of order $\sqrt{\kappa t}$; it becomes a narrow spike at zero and converges to the delta distribution. Thus the convolution tends to $u_0(x)$.