Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 24, pp. 1-10.

Title: Almost optimal local well-posedness for modified Boussinesq equations

Authors: Dan-Andrei Geba (Univ. of Rochester, Rochester, USA)
         Bai Lin         (Univ. of Rochester, Rochester, USA)

Abstract:
 In this article, we investigate a class of modified Boussinesq equations,
 for which we provide first an alternate proof of local well-posedness in
 the space  $(H^s\cap L^\infty)\times (H^s\cap L^\infty)(\mathbb{R})$ ($s\geq 0$)
 to the one obtained  by Constantin and Molinet [7].
 Secondly, we show that the associated flow map is not smooth when considered
 from $H^s\times H^s(\mathbb{R})$ into $H^s(\mathbb{R})$ for s<0, 
 thus providing a threshold for  the regularity needed to perform a Picard 
 iteration for these equations.
 
Submitted June 12, 2019. Published March 19, 2020.
Math Subject Classifications: 35B30, 35Q55.
Key Words: Modified Boussinesq equation; well-posedness; ill-posedness.