(a) Show that the Cauchy problem for $u(x, t)$ satisfying

$$u_{t}+u=u_{x x}$$

with initial data $u(x, 0)=u_{0}(x)$, which is a smooth $2 \pi$-periodic function of $x$, defines a strongly continuous one parameter semi-group of contractions on the Sobolev space $H_{\text {per }}^{s}$ for any $s \in\{0,1,2, \ldots\}$.

(b) Solve the Cauchy problem for the equation

$$u_{t t}+u_{t}+\frac{1}{4} u=u_{x x}$$

with $u(x, 0)=u_{0}(x), u_{t}(x, 0)=u_{1}(x)$, where $u_{0}, u_{1}$ are smooth $2 \pi$-periodic functions of $x$, and show that the solution is smooth. Prove from first principles that the solution satisfies the property of finite propagation speed.

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

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

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