In this question, functions are all real-valued, and

$$H_{p e r}^{s}=\left\{u=\sum_{m \in \mathbb{Z}} \hat{u}(m) e^{i m x} \in L^{2}:\|u\|_{H^{s}}^{2}=\sum_{m \in \mathbb{Z}}\left(1+m^{2}\right)^{s}|\hat{u}(m)|^{2}<\infty\right\}$$

are the Sobolev spaces of functions $2 \pi$-periodic in $x$, for $s=0,1,2, \ldots$

State Parseval's theorem. For $s=0,1$ prove that the norm $\|u\|_{H^{s}}$ is equivalent to the norm ||$_{s}$ defined by

$$\|u\|_{s}^{2}=\sum_{r=0}^{s} \int_{-\pi}^{+\pi}\left(\partial_{x}^{r} u\right)^{2} d x$$

Consider the Cauchy problem

$$u_{t}-u_{x x}=f, \quad u(x, 0)=u_{0}(x), \quad t \geqslant 0,$$

where $f=f(x, t)$ is a smooth function which is $2 \pi$-periodic in $x$, and the initial value $u_{0}$ is also smooth and $2 \pi$-periodic. Prove that if $u$ is a smooth solution which is $2 \pi$-periodic in $x$, then it satisfies

$$\int_{0}^{T}\left(u_{t}^{2}+u_{x x}^{2}\right) d t \leqslant C\left(\left\|u_{0}\right\|_{H^{1}}^{2}+\int_{0}^{T} \int_{-\pi}^{\pi}|f(x, t)|^{2} d x d t\right)$$

for some number $C>0$ which does not depend on $u$ or $f$.

State the Lax-Milgram lemma. Prove, using the Lax-Milgram lemma, that if

$$f(x, t)=e^{\lambda t} g(x)$$

with $g \in H_{p e r}^{0}$ and $\lambda>0$, then there exists a weak solution to (1) of the form $u(x, t)=e^{\lambda t} \phi(x)$ with $\phi \in H_{\text {per. }}^{1}$. Does the same hold for all $\lambda \in \mathbb{R}$ ? Briefly explain your answer.