On the minimal Blow-up rate for the 2D generalized Zakharov- Kuznetsov model

Abstract

In this note we consider the generalized Zakharov-Kuznetsov equation in $\mathbb R^2$, for initial conditions in the Sobolev space $H^s$ with $s>3/4.$ Assuming that there is a blow-up solution at finite time $T^{*}$, we obtain a lower bound for the blow-up rate of that solution, expressed in terms of a lower bound for the $H^s$ norm of the solution. In the particular case of the modified Zakharov-Kuznetsov equation, {\color{teal} a nontrivial gap is found between conjectured blow-up rates and our results.} The analysis is based on properly quantifying the linear estimates given by Faminskii \cite{Faminskii}, as well as the local well-posedness theory of Linares and Pastor \cite{Linares2009,LinaresPastor}, combined with an argument developed by Weissler \cite{Weissler} and {\color{teal} Colliander, Czuback and Sulem} \cite{Colliander} in the context of the semilinear heat equations.