Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 319 / 1 / f / Solution

Write the nonlinearity asIn one dimension, the Sobolev embedding theorem gives $H^1(\mathbb{T})\hookrightarrow L^{\infty }(\mathbb{T})$. HenceThus $N:\mathcal{H}\to \mathcal{H}$ is locally Lipschitz.

On $C([0,T];\mathcal{H})$ defineOn a ball of radius $R$, unitarity givesChoose $R>\Vert Z_0{\Vert }_{\mathcal{H}}$ and then $T>0$ small enough that the first bound preserves the ball and $CTR<1$. The contraction mapping theorem gives a unique fixed point. Precisely, the local mild solution is

For general energy data $Z_0\in \mathcal{H}$, this solution need not be differentiable in $\mathcal{H}$. If $Z_0\in D(A)=H^2\times H^1$, standard semilinear evolution theory and the smoothness of $N$ give a local classical solution.

The blow-up alternative for a semilinear evolution equation says that the solution continues while its $\mathcal{H}$ norm stays finite, but global existence does not hold for every datum. Spatially constant solutions obeyFor sufficiently large $y(0)>1$ with $y^{\prime }(0)\ge 0$, the solution grows until $y^{\prime \prime }\ge y^2/2$ and blows up in finite time. These constant functions are periodic and belong to every Sobolev space, so they provide finite-time blow-up examples for the original equation.