Suppose we have $$u_t + f(u) u_x = 0$$ where $t, x > 0$, and initial conditions $u(x,0) = C$, where $C>0$ is constant, and $u(0,t) = g(t)$, where $t>0$. We know the solution is $$u(x,t) = F(x-f(u) t )$$ for any differentiable $F$, and characteristics are given by $x - f(u)t = r $. I am trying to find where shocks will form and find the solution in such case.
However, I don't quite understand the problem since $u(x,0) = F(x) = C$ and so $$u(x,t) = F( x - ct) = C $$ so the solution is a constant. What is my mistake here?