State the Cauchy-Kovalevskaya theorem, including a definition of the term noncharacteristic.

For which values of the real number $a$, and for which functions $f$, does the CauchyKovalevskaya theorem ensure that the Cauchy problem

$$u_{t t}=u_{x x}+a u_{x x x x}, \quad u(x, 0)=0, u_{t}(x, 0)=f(x)$$

has a local solution?

Now consider the Cauchy problem (1) in the case that $f(x)=\sum_{m \in \mathbb{Z}} \hat{f}(m) e^{i m x}$ is a smooth $2 \pi$-periodic function.

(i) Show that if $a \leqslant 0$ there exists a unique smooth solution $u$ for all times, and show that for all $T \geqslant 0$ there exists a number $C=C(T)>0$, independent of $f$, such that

$$\int_{-\pi}^{+\pi}|u(x, t)|^{2} d x \leqslant C \int_{-\pi}^{+\pi}|f(x)|^{2} d x$$

for all $t:|t| \leqslant T$.

(ii) If $a=1$ does there exist a choice of $C=C(T)$ for which (2) holds? Give a full justification for your answer.