(a) Define characteristic hypersurfaces and state a local existence and uniqueness theorem for a quasilinear partial differential equation with data on a non-characteristic hypersurface.

(b) Consider the initial value problem

$$3 u_{x}+u_{y}=-y u, \quad u(x, 0)=f(x),$$

for a function $u: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $C^{1}$ initial data $f$ given for $y=0$. Obtain a formula for the solution by the method of characteristics and deduce that a $C^{1}$ solution exists for all $(x, y) \in \mathbb{R}^{2}$.

Derive the following (well-posedness) property for solutions $u(x, y)$ and $v(x, y)$ corresponding to data $u(x, 0)=f(x)$ and $v(x, 0)=g(x)$ respectively:

$$\sup _{x}|u(x, y)-v(x, y)| \leqslant \sup _{x}|f(x)-g(x)| \quad \text { for all } y .$$

(c) Consider the initial value problem

$$3 u_{x}+u_{y}=u^{2}, \quad u(x, 0)=f(x),$$

for a function $u: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $C^{1}$ initial data $f$ given for $y=0$. Obtain a formula for the solution by the method of characteristics and hence show that if $f(x)<0$ for all $x$, then the solution exists for all $y>0$. Show also that if there exists $x_{0}$ with $f\left(x_{0}\right)>0$, then the solution does not exist for all $y>0$.