Regularity Properties, Representation of Solutions, and Spectral Asymptotics of Systems with Multiplicities

Properties of solutions of generic hyperbolic systems with multiple characteristics with microlocally diagonalizable principal part are investigated. Solutions are represented as a Picard series with terms in the form of iterated Fourier integral operators. It is shown that this series is an asymptotic expansion with respect to smoothness under quite general geometric conditions on characteristics. Both constant and variable multiplicities are allowed. Propagation of singularities is described and sharp regularity properties of solutions are obtained. Results are applied to establish regularity estimates for scalar weakly hyperbolic equations with involutive characteristics. They are also applied to derive the first and second terms of the spectral asymptotics for the corresponding self-adjoint elliptic systems.