(i) Consider the problem of solving the equation

$$\sum_{j=1}^{n} a_{j}(\mathbf{x}) \frac{\partial u}{\partial x_{j}}=b(\mathbf{x}, u)$$

for a $C^{1}$ function $u=u(\mathbf{x})=u\left(x_{1}, \ldots, x_{n}\right)$, with data specified on a $C^{1}$ hypersurface $\mathcal{S} \subset \mathbb{R}^{n}$

$$u(\mathbf{x})=\phi(\mathbf{x}), \quad \forall \mathbf{x} \in \mathcal{S}$$

Assume that $a_{1}, \ldots, a_{n}, \phi, b$ are $C^{1}$ functions. Define the characteristic curves and explain what it means for the non-characteristic condition to hold at a point on $\mathcal{S}$. State a local existence and uniqueness theorem for the problem.

(ii) Consider the case $n=2$ and the equation

$$\frac{\partial u}{\partial x_{1}}-\frac{\partial u}{\partial x_{2}}=x_{2} u$$

with data $u\left(x_{1}, 0\right)=\phi\left(x_{1}, 0\right)=f\left(x_{1}\right)$ specified on the axis $\left\{\mathbf{x} \in \mathbb{R}^{2}: x_{2}=0\right\}$. Obtain a formula for the solution.

(iii) Consider next the case $n=2$ and the equation

$$\frac{\partial u}{\partial x_{1}}-\frac{\partial u}{\partial x_{2}}=0$$

with data $u(\mathbf{g}(s))=\phi(\mathbf{g}(s))=f(s)$ specified on the hypersurface $\mathcal{S}$, which is given parametrically as $\mathcal{S} \equiv\left\{\mathbf{x} \in \mathbb{R}^{2}: \mathbf{x}=\mathbf{g}(s)\right\}$ where $\mathbf{g}: \mathbb{R} \rightarrow \mathbb{R}^{2}$ is defined by

$$\begin{aligned} &\mathbf{g}(s)=(s, 0), \quad s<0 \\ &\mathbf{g}(s)=\left(s, s^{2}\right), \quad s \geqslant 0 \end{aligned}$$

Find the solution $u$ and show that it is a global solution. (Here "global" means $u$ is $C^{1}$ on all of $\mathbb{R}^{2}$.)

(iv) Consider next the equation

$$\frac{\partial u}{\partial x_{1}}+\frac{\partial u}{\partial x_{2}}=0$$

to be solved with the same data given on the same hypersurface as in (iii). Explain, with reference to the characteristic curves, why there is generally no global $C^{1}$ solution. Discuss the existence of local solutions defined in some neighbourhood of a given point $\mathbf{y} \in \mathcal{S}$ for various $\mathbf{y}$. [You need not give formulae for the solutions.]