= Solution
The <inverse function theorem> applied to $X_t$ makes $u_0\circ X_t^{-1}$ smooth. Since $u_0(\pm A)=0$, $X_t(\pm A)=\pm A+ct$. Monotonicity of the <characteristic flow map> and the vanishing of $u_0$ outside $[-A,A]$ imply
$$
\boxed{\operatorname{supp}u(t,\cdot)\subset[-A+ct,A+ct]\quad(0<t<t^*).}
$$
Thus the solution is a <smooth function> with compact support at each such time. The fact that some interior <characteristic speeds> differ from $c$ causes no difficulty: they cannot cross the two exterior <characteristic curves> before $t^*$.
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