We study hyperbolic systems of one - dimensional partial differential equations under general , possibly non-local boundary conditions. A large class of evolution equations, either on individual 1- dimensional intervals or on general networks , can be reformulated in our rather flexible formalism , which generalizes the classical technique of first - order reduction . We study forward and backward well - posedness ; furthermore , we provide necessary and sufficient conditions on both the

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... y conditions and the coefficients arising in the first - order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied . p, li { white-space: pre-wrap; }