Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 1 / b

The wave operator has principal symbolThe conormal to the initial plane is $dt$, and $p(dt)=-1\ne 0$. The plane is therefore non-characteristic.

The principal symbol is ${\tau }^2+|\xi {|}^2$, so its value on $dt$ is $1$. The initial plane is non-characteristic.

When the heat equation is viewed as a second-order equation, its principal symbol contains only the spatial second derivatives:up to an irrelevant sign. It vanishes on $dt$, so the initial plane is characteristic in this second-order sense. The equation remains a well-posed first-order evolution equation in time; these are different notions of order.

The principal symbol isIts value on $dt$ is $1$, independently of the data, so the initial plane is non-characteristic everywhere.

For $\varphi (x)=|x{|}^2-1$, the conormal on the unit sphere is $d\varphi =(0,2x)$. Henceon the sphere. It is non-characteristic at every point.

For $\varphi =t-x_1$, the conormal is $dt-d{x}_1$, andThis null hyperplane is a characteristic hypersurface of the wave equation.

The second-order principal symbol of the heat operator evaluated on $dt-d{x}_1$ is $-1$, up to the overall sign convention. Thus this tilted hypersurface is non-characteristic.