For hyperbolic PDE's, we have two aspects that distinguish them from elliptic and parabolic PDE's:
 
 - Usually the loss of derivatives is worse than in the elliptic case - in fact, even worse than in the more general subelliptic case. Typically, you lose a full (weak = Sobolev) derivative in *a priori* regularity estimates for linear hyperbolic PDE's: the so-called *energy estimates*. This loss is essentially sharp and can be tracked back to the fundamental theorem of Calculus. As pointed by Willie Wong in his answer, this can be refined if you take into account the characteristic set of the principal symbol of the operator, by means of techniques collectively called *microlocal analysis*. More precisely, worse than elliptic-type regularity must propagate along the bicharacteristic curves of the principal symbol, which are nonexistent in the elliptic case.
 - Even if you account for the above loss of derivatives, the "optimal" relative shape of the regions $B_1,B_2$ is different due to a hallmark property of hyperbolic PDE's - namely, the *finite speed of propagation* of (perturbations of) solutions. Typically, $B_2$ will be a "cone-shaped" shadow region cast by $B_1$, whose precise form is once again dictated by the principal symbol of the operator.