I am finding an analytic expression for the solution of the transport PDE: $$u_t+\left(\frac{1-2u(x,t)}{a}\right)u_x = 0,\quad a= \text{const.}, \quad x>0, \quad t >0$$ $$u(x=0,t) = u_0, \quad u_0 = \text{const.} \in (0,1)$$ $$u(x,t=0) = \varphi(x)$$
I have got the solution:
$$ u(x,t) = \varphi\left(x-\frac{1-2u(x,t)}{a}\right) $$
My question is how to use boundary condition $u_0$? I have checked many different books and articles, but stuck though.
Thank you.
Correct. The characteristic curves starting from $(x,t) = (x_0, 0)$ are the lines $x = x_0 + (1-2u)t/a$ along which $$u=\varphi(x_0) = \varphi\!\left(x - \frac{1-2u}{a}t\right)$$ is constant. In particular, the line starting from $x_0 = 0^+$ satisfies $x = (1-2\varphi(0))t/a$. Note that the implicit expression above is only valid up to the breaking time.
For the other part, characteristic curves starting from $(x,t) = (0, t_0)$ are the lines $t = t_0 + ax/(1-2u)$ along which $u = u_0$ is constant. In particular, the line starting from $t_0 = 0^+$ satisfies $t = ax/(1-2u_0)$. Now, it remains to analyze the interaction of both parts... Do these curves intersect, or do they separate? What if $\varphi(0)=u_0$? (Hint: sketch the characteristics in the $x$-$t$ plane)