Various plasma phenomena are mathematically studied using a fundamental fluid model for plasmas, called the Euler-Poisson system. Among them, plasma solitary waves are of our interest, for which existence, stability, and the time-asymptotic behavior of the solitary wave will be briefly discussed. On the other hand, to study nonlinear stability, a question of existence of smooth global solution naturally arises, which is completely open, to the best of our knowledge. We introduce the finite-time blow-up results for the Euler-Poisson system, and discuss the related open questions. This talk is based on joint work with Junsik Bae at UNIST.