Domain of definition for $u_x + uu_y = 1$ How do i find the domain of definition for $u_x + uu_y = 1$
with $u = x/2$ on $y=x$ , $0 \leq x \leq 1$ 
I parametrise by letting $y=s$ , $x=s$ , $u=s/2$ , $0 \leq s \leq 1$ at $t=0$
The characteristic equations are: 
$dx/dt = 1$, $dy/dt = u$, $du/dt = 1$
Solving $dx/dt = 1$ gives $x=t +s$
Solving $du/dt = 1$ gives $u=t+ s/2$
Solving $dy/dt = u$ gives $y = (1/2)t^2 + st/2 + s$
This is where I get stuck,
$t=x-s$ so $y = (1/2)(x-s)^2 + (1/2)s(x-s) + s$
and $0 \leq s \leq 1$ so am i correct in saying that the domain of definition is the region between $y = (1/2)x^2 $ and $y = (1/2)(x-1)^2 + (1/2)(x-1) + 1$

$$u_x+uu_y= \tag 1$$
Your three equations written on a equivalent form:
$$\frac{dx}{1}=\frac{dy}{u}=\frac{du}{1}=dt$$
A first family of characteristic equations comes from $\frac{dx}{1}=\frac{du}{1}$
$$u-x=c_1$$
A second family of characteristic equations comes from $\frac{dy}{u}=\frac{du}{1}$
$$\frac{u^2}{2}-y=c_2$$
The general solution of the PDE expressed on the form of implicite equation is :
$$\frac{u^2}{2}-y=F(u-x) \tag 2$$
where $F$ is an arbitrary equation, to be determined according to boundary condition :
$$u(x,x)=\frac{x}{2}\quad\implies\quad\frac{x^2}{8}-x=F\left(\frac{x}{2}-x\right)$$
$X=-\frac{x}{2}\quad;\quad x=-2X$
$$F(X)=\frac{(-2X)^2}{8}-(-2X)=\frac{X^2}{2}+2X$$
So, the function $F(X)$ is determined. we put it into the general solution Eq.$(2)$ where $X=u-x$
$$\frac{u^2}{2}-y=\frac{(u-x)^2}{2}+2(u-x)$$
After simplification :
$$u(x,y)=\frac{y+\frac{x^2}{2}-2x}{x-2}$$
You can find the domain of definition.
