Nonstrictly hyperbolic systems and their application to the study of Euler–Poisson equations

We consider inhomogeneous non-strictly hyperbolic systems of two equations, which are a formal generalization of transformed one-dimensional Euler–Poisson equations. For such systems, a complete classification of the behavior of solutions is carried out depending on the right-hand side. We found a criteria for formation of singularities in solutions of the Cauchy problem in terms of initial data. We determine domains of attraction of equilibria for the extended system for derivatives. We prove existence of solutions in a form of simple waves. The results obtained are applied to the study of main model cases of Euler–Poisson equations.