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

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.