= Solution
For any fixed $t>0$, the characteristic map $X_t(\xi)=\xi+t u_I(\xi)$ has <derivative> $1+t u_I'(\xi)>1$. It is strictly increasing and onto: for $\xi\geq0$ it is at least $\xi+t u_I(0)$, and for $\xi\leq0$ 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 gives
$$
u_x=\frac{u_I'(\xi)}{1+t u_I'(\xi)},\qquad
u_t=-\frac{u_I(\xi)u_I'(\xi)}{1+t u_I'(\xi)}=-uu_x.
$$
The <derivatives> are continuous, and $u_x\leq1/t$ for positive times. Thus \b[a classical solution exists for all $t>0$], without any characteristic intersection or shock formation.
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