Let $C_{p e r}^{\infty}=\left\{u \in C^{\infty}(\mathbb{R}): u(x+2 \pi)=u(x)\right\}$ be the space of smooth $2 \pi$-periodic functions of one variable.

(i) For $f \in C_{p e r}^{\infty}$ show that there exists a unique $u_{f} \in C_{p e r}^{\infty}$ such that

$$-\frac{d^{2} u_{f}}{d x^{2}}+u_{f}=f$$

(ii) Show that $I_{f}\left[u_{f}+\phi\right]>I_{f}\left[u_{f}\right]$ for every $\phi \in C_{p e r}^{\infty}$ which is not identically zero, where $I_{f}: C_{p e r}^{\infty} \rightarrow \mathbb{R}$ is defined by

$$I_{f}[u]=\frac{1}{2} \int_{-\pi}^{+\pi}\left[\left(\frac{\partial u}{\partial x}\right)^{2}+u^{2}-2 f(x) u\right] d x$$

(iii) Show that the equation

$$\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}+u=f(x)$$

with initial data $u(0, x)=u_{0}(x) \in C_{\text {per }}^{\infty}$ has, for $t>0$, a smooth solution $u(t, x)$ such that $u(t, \cdot) \in C_{p e r}^{\infty}$ for each fixed $t>0$. Give a representation of this solution as a Fourier series in $x$. Calculate $\lim _{t \rightarrow+\infty} u(t, x)$ and comment on your answer in relation to (i).

(iv) Show that $I_{f}[u(t, \cdot)] \leqslant I_{f}[u(s, \cdot)]$ for $t>s>0$, and that $I_{f}[u(t, \cdot)] \rightarrow I_{f}\left[u_{f}\right]$ as $t \rightarrow+\infty$.