  In this paper, we consider the Cauchy problem for the Hunter-Saxton (HS)
equation on the line. Firstly, we establish the local well-posedness for the
integral form of the (HS) equation by constructing some special spaces
$E^s_{p,r}$, which mix Lebesgue spaces and homogeneous Besov spaces. Then we
present a global existence result and provide a sufficient condition for strong
solutions to blow up in finite time for the equation. Finally, we give the
ill-posedness and the unique continuation of the Hunter-Saxton equation.
