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

TL;DR

This paper investigates the minimal blow-up rate for solutions to the 2D generalized Zakharov-Kuznetsov equation, establishing lower bounds and highlighting gaps in the modified case.

Contribution

It provides a lower bound for blow-up rates in the 2D generalized Zakharov-Kuznetsov equation and identifies a gap between conjectured and proven rates for the modified version.

Findings

Lower bound for blow-up rate in Sobolev space $H^s$, $s>3/4$

Identification of a gap in the modified Zakharov-Kuznetsov case

Quantification of linear estimates and well-posedness results

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.