 Solution

For any fixed $t>0$, the characteristic map $X_t(\xi )=\xi +t{u}_I(\xi )$ has derivative $1+t{u}_I^{\prime }(\xi )>1$. It is strictly increasing and onto: for $\xi \ge 0$ it is at least $\xi +t{u}_I(0)$, and for $\xi \le 0$ at most that quantity. Hence the inverse function theorem supplies a global $C^1$ inverse. Define $u(x,t)=u_I(X_t^{-1}(x))$. Differentiating givesThe derivatives are continuous, and $u_x\le 1/t$ for positive times. Thus a classical solution exists for all $t>0$, without any characteristic intersection or shock formation.