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$\{\begin{array}{cc}{\partial }_{t}u+{\partial }_x\Delta u+\mu {\partial }_x{u}^{k+1}=0,& (x,y)\in R^2,t\in R,\\ u(x,y,0)=u_0(x,y),\end{array}$ 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 complex strip in $C^2$ and have radius of spatial analyticity ${\sigma }_0$. For both $k=1$ and $k=2$, considering a symmetrized version, 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 $c{T}^{-(4+ϵ)}$, $ϵ>0$, $c>0$. 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 $c{T}^{-\frac{4}{3}}$, $c>0$, 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 [41] and by Quian and Shan in [33] where the authors have obtained lower bounds involving exponential decay.