Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 14B / i / Solution

For $a=0$ and $\kappa >0$, the heat kernel givesAs $t\downarrow 0$, $K_t$ 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 $u_0(x)$.