 Solution

The solution is constant on characteristics $x_1+x_2=\eta$. On the initial curve this invariant is $h(s)=s$ for $s<0$ and $h(s)=s+s^2$ for $s\ge 0$. This is a $C^1$ bijection of $\mathbb{R}$, with everywhere positive derivative. Its inverse isTherefore $\boxed{u(x_1,x_2)=f(h^{-1}(x_1+x_2)).}$ The inverse is $C^1$ even at zero, so for $C^1$ data this is a global $C^1$ solution. Every characteristic meets the initial curve exactly once.