The Cauchy initial value problem on different surfaces (Q1374941)

Solutions to a quasilinear system of partial differential equations are considered when the initial values for different functions are given on different surfaces. Sufficient conditions for the existence and uniqueness of the solutions to the stated problem are found. Counterexamples are given showing that if certain sufficient conditions are broken, then there does not exist an analytical solution. It is proved that the question of the existence of a piecewise analytical solution to the problem of sharply pushing a piston into a gas reduces to the considered Cauchy problem with initial values given on different surfaces.