Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 14B (Methods)

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 $u_0(x)$.

When $\kappa =0$, the characteristics satisfy $x-at=\text{constant}$ and $u$ is constant along them. Therefore

Set $v(t,z)=u(t,z+at)$. Then $v_t=\kappa v_{zz}$ and $v(0,z)=u_0(z)$. Applying the heat kernel and returning to $x$ gives the advection-diffusion heat-kernel solution