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. First we discuss existence, stability, and the time-asymptotic behavior of the solitary wave. To study nonlinear stability, a question of existence of smooth global solution naturally arises, which is completely open, to the best of our knowledge. For evidence of a negative answer, we introduce the finite-time blow-up results for the Euler-Poisson system, and discuss the related open questions. This is joint work with J. Bae (UNIST) and J. Choi (SKKU).

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