Write the nonlinearity as
$$
N(u,v)=(0,u^2).
$$
In one dimension, the <Sobolev embedding theorem> gives $H^1(\mathbb T)\hookrightarrow L^\infty(\mathbb T)$. Hence
$$
\|u^2-w^2\|_{L^2}
\leq\|u+w\|_{L^\infty}\|u-w\|_{L^2}
\leq C(\|u\|_{H^1}+\|w\|_{H^1})\|u-w\|_{H^1}.
$$
Thus $N:\mathcal H\to\mathcal H$ is locally Lipschitz.

On $C([0,T];\mathcal H)$ define
$$
(\Phi Z)(t)=U(t)Z_0+int_0^tU(t-s)N(Z(s))\,ds.
$$
On a ball of radius $R$, unitarity gives
$$
\|\Phi Z\|_\infty\leq\|Z_0\|_{\mathcal H}+CTR^2,
$$
$$
\|\Phi Z-\Phi W\|_\infty\leq CTR\|Z-W\|_\infty.
$$
Choose $R>\|Z_0\|_{\mathcal H}$ 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
$$
\boxed{
Z\in C([0,T];\mathcal H),
\qquad
Z(t)=U(t)Z_0+int_0^tU(t-s)(0,u(s)^2)\,ds}.
$$

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 obey
$$
y''+y=y^2.
$$
For sufficiently large $y(0)>1$ with $y'(0)\geq0$, the solution grows until $y''\geq y^2/2$ 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.

Solution

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

For a locally Lipschitz semilinear evolution, a maximal mild solution either exists for all positive time or its phase-space norm becomes unbounded as the finite maximal time is approached.


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