Well-posedness issues for the generalized Benjamin--Bona--Mahony equation

In this paper, we consider the one-dimensional generalized Benjamin--Bona--Mahony (gBBM) equation $$(1-\partial_x^2)u_t+(u+u^p)_x=0,\qquad p=2,3,4,\dots,$$ posed either on the real line $\mathbb R$ or on the torus $\mathbb T$. This equation may be viewed as a regularized model for the propagation of long-crested surface water waves. The main results of this work are threefold: \medskip First, we establish \emph{unconditional local well-posedness} in the class $C([0,T];H^s)$ without imposing any auxiliary spaces for $$s\ge \frac{p-2}{2p},$$ which is \emph{sharp} in the sense that the multilinear estimate in $H^s$ is optimal. In addition, we prove \emph{unconditional uniqueness} for all distributional solutions in $L^\infty((0,T);H^s)$. \medskip Second, we show that below this regularity threshold, the flow map cannot be of class $C^p$. Precisely, if the flow map is well-defined and continuous near the origin from $H^s$ to $C([0,T];H^s)$ for every $s<\frac{p-2}{2p}$, then it cannot be of class $C^p$ at the origin. The proof is based on a high-to-low frequency interaction, implemented differently on $\mathbb R$ and $\mathbb T$. \medskip Third, in the odd-power case, we prove \emph{global well-posedness} below $H^1$ in the following cases: $p=3$ with $s\ge \frac14$, and $p=5$ with $s>\frac12$. To the best of our knowledge, these are the first global well-posedness results in the Sobolev framework for the generalized BBM equation below $H^1$. The argument is based on the Bona--Tzvetkov approach \cite{BT}, while being initially inspired by Bourgain's high--low method \cite{Bourgain1998, Bourgain1999}. A key new ingredient is the use of a Hamiltonian conservation law below the $H^1$ energy level. This allows us to control the higher-degree nonlinear contributions in the energy estimate, thereby preventing the Gr\"onwall iteration from blowing up.