On the algebraic lower bound for the radius of spatial analyticity for the Zakharov-Kuznetsov and modified Zakharov-Kuznetsov equations

TL;DR

This paper establishes algebraic lower bounds on the decay of the radius of spatial analyticity for solutions to the 2D Zakharov-Kuznetsov equations, improving previous exponential decay bounds.

Contribution

It proves that the radius of spatial analyticity cannot decay faster than a specific algebraic rate over time for both the standard and modified ZK equations.

Findings

Radius of analyticity remains constant over a short time interval.

Lower bounds on decay rate are algebraic, not exponential.

Results improve previous bounds by Shan, Zhang, Quian, and Shan.

Abstract

We consider the initial value problem (IVP) for the 2D generalized Zakharov-Kuznetsov (ZK) equation \begin{equation} \begin{cases} \partial_{t}u+\partial_{x}\Delta u+\mu \partial_{x}u^{k+1}=0, \,\;\; (x, y) \in \mathbb{R}^2, \, t \in \mathbb{R},\\ u(x,y,0)=u_0(x,y), \end{cases} \end{equation} where $\Delta=\partial_x^2+\partial_y^2$, $\mu=\pm 1$, $k=1,2$ and the initial data $u_0$ is real analytic in a strip around the $x$-axis of the complex plane and have radius of spatial analyticity $\sigma_0$. For both $k=1$ and $k=2$ we prove that there exists $T_0>0$ such that the radius of spatial analyticity of the solution remains the same in the time interval $[-T_0, T_0]$. We also consider the evolution of the radius of spatial analyticity when the local solution extends globally in time. For the Zakharov-Kuznetsov equation ($k=1$), we prove that, in both focusing ($\mu=1$) and defocusing ($\mu=-1$) cases, and for any $T> T_0$, the radius of analyticity cannot decay faster than $cT^{-4+\epsilon}$, $\epsilon>0$, $c>0$. For the modified Zakharov-Kuznetsov equation ($k=2)$ in the defocusing case ($\mu=-1$), we prove that the radius of spatial analyticity cannot decay faster than $cT^{-\frac{4}{3}}$, $c>0$, for any $T>T_0$. These results on the algebraic lower bounds for the evolution of the radius of analyticity improve the ones obtained by Shan and Zhang in [J. Math. Anal. Appl., 501 (2021) 125218] and by Quian and Shan in [Nonlinear Analysis, 235 (2023) 113344] where the authors have obtained lower bounds involving exponential decay.