Blowup Solutions of the “Bad” Boussinesq Equation

Abstract

Abstract. We study blowup solutions of the “bad” Boussinesq equation and prove that a wide range of asymptotic scenarios can happen. For example, for each [Formula: see text], [Formula: see text] and [Formula: see text], we prove that there exist Schwartz class solutions [Formula: see text] on [Formula: see text] such that [Formula: see text] and [Formula: see text] as [Formula: see text]. We also prove that for any [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], there exist Schwartz class solutions [Formula: see text] on [Formula: see text] such that [Formula: see text] for each [Formula: see text] such that [Formula: see text]; [Formula: see text] for each [Formula: see text] such that [Formula: see text]; and [Formula: see text] as [Formula: see text] for each [Formula: see text] such that [Formula: see text]. In particular, when [Formula: see text], this result establishes the existence of wave-breaking solutions, i.e., solutions that remain bounded but whose [Formula: see text]-derivative blows up in finite time.