Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 23I / c / Solution

Taking the Fourier transform of the equation in the sense of distributions givesThe Fourier multiplieris everywhere positive. Define $\hat{u}=\hat{f}/m$. Sincewe haveThus $u\in H^{s+4}$ exists and obeys the required estimate. If two solutions existed, their difference would have $m\hat{u}=0$; positivity of $m$ proves uniqueness.

A classical solution needs continuous derivatives through order four. Applying the Sobolev embedding theorem with $k=4$ to $u\in H^{s+4}$ shows that this is guaranteed whenor equivalently