Summary: We consider an initial value problem for nonlinear partial differential equations describing the motion of inhomogeneous anisotropic hyperelastic medium. We prove the theorems about the existence (locally in time) and uniqueness of a smooth solution to this initial valve problem. In order to do it, we apply the modified Sommerfeld method to convert the considered initial value problem into a first-order quasilinear symmetric hyperbolic initial value problem. Next, we prove the blow-up at finite time of the solution of the above-mentioned initial problem under some assumption about the stored-energy function of hyperelastic materials.