On the local ill-posedness of the generalized \(p\)-Gardner equation. (Q2343878)

This paper deals with the initial value problem (IVP) for a generalization of the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations, called the generalized Gardner equation. Previous work on this equation has been done by \textit{M. A. Alejo} (see ref. [J. Math. Anal. Appl. 396, No. 1, 256--260 (2012; Zbl 1256.35197)] and [NoDEA, Nonlinear Differ. Equ. Appl. 19, No. 4, 503--520 (2012; Zbl 1254.35199)]). In particular, the local well-posedness and the lack of a uniformly continuous solution to the IVP was proved, depending on the regularity of the Sobolev space \(H^s\) used; the threshold is the same as in the mKdV equation: \(s=-\frac{1}{4}\). In the paper under review, the authors use the method developed by Birnir, Ponce and Svanstedt to show the strong ill-posedness of the IVP for the Gardner equation below \(s=-\frac{1}{2}\), the regularity corresponding to the ``formal scaling'' of the equation. Their proof uses the fact that solitary waves have finite \(H^s\) norm, \(s<-\frac{1}{2}\), independently of the scaling, as well as the delta function, which is the strong \(H^s\) limit of solitary waves as the scaling converges to infinity.