Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 30C (Partial Differential Equations)

Choose the Fourier transform conventionIt maps the Schwartz space to itself. For a tempered distribution $u$, define the transform by bilinear distributional duality:For an ordinary integrable function this agrees with the first formula by interchanging its integrals. The absence of complex conjugation is the distribution-pairing convention.

Rotate the polar axis to $\xi$. Since $W_t$ is a finite sphere measure, its Fourier transform iswith value $t$ at $\xi =0$. Fourier transforming the wave equation gives ${\hat{u}}_{tt}+|\xi {|}^2\hat{u}=0$ with initial displacement zero and velocity $\hat{g}$. Its solution is this multiplier times $\hat{g}$, so inversion gives $u=W_t\ast g$. Thus the Kirchhoff formula isThis extends to arbitrary smooth $g$, without any growth restriction: for each finite spacetime neighborhood, multiply $g$ by a compactly supported smooth cutoff equal to one on every sphere occurring in that neighborhood. The formula is unchanged there, so its verified differential equation remains valid locally. Its initial values follow from $u(t,x)=tg(x)+o(t)$ and differentiation under the sphere integral. Since the neighborhood was arbitrary, it is a smooth solution everywhere for $t\ge 0$.

Let $w$ be the difference between a $C^2$ solution and the formula in part (ii). It satisfies $w_{tt}-\Delta w=0$ and zero initial data. Fix $(T,x_0)$ and $\epsilon >0$, and put $B_t=B(x_0,T+\epsilon -t)$. Define the local energyDifferentiating on the shrinking ball and integrating by parts givessince $2w_t{\partial }_{n}w\le w_t^2+({\partial }_{n}w{)}^2\le w_t^2+|\nabla w{|}^2$. Initially $E=0$, hence $w_t=\nabla w=0$ inside each ball. In particular $w(t,x_0)$ is constant from its initial value zero, so $w(T,x_0)=0$. Since the point was arbitrary, every $C^2$ solution equals the Kirchhoff solution. This shrinking cone energy argument uses only energy on compact sets, so it does not assume global decay of the competing solution.

For the difference of two solutions the equation is $w_{tt}+w_t-\Delta w=0$ with zero initial data. The same shrinking-ball energy now obeysAgain $E(0)=0$ forces the difference to vanish at every chosen spacetime point. Therefore the two $C^2$ solutions agree for every $t>0$. Damping contributes an additional negative energy term and does not weaken the local uniqueness proof.