Boundary value advection equation

Question

Given the signalling problem from the exercises of Chapter 2.1 in the Introduction to Non-Linear PDE's by Logan we have \begin{equation} \partial_t u - x^2 \cdot \partial_x u= 0 \\ u(0,t) = g(t) \end{equation} Using the method of characteristics we get $$ \frac{\text{d} u}{\text{d} t} = 0 \text{ along } x = \frac{1}{t+\eta} = C(t, \eta) $$ for some $\eta \in \mathbb{R}$. However I'm not sure how to proceed to get a general solution for $u$. I'd assume we have to separate the characteristics with different solutions for $x > C(t, \eta)$ and $x < C(t, \eta)$.