Improved Gevrey Class Regularity of the Kadomtsev Petviashvili Equation
Abstract
In this paper, we improve and extend the results obtained by Boukarou et al. \cite{boukarou1} on the Gevrey regularity of solutions to a fifth-order Kadomtsev-Petviashvili (KP)-type equation. We establish Gevrey regularity in the time variable for solutions in $2+1$ dimensions, providing a sharper result obtained through a new analytical approach. Assuming that the initial data are Gevrey regular of order $\sigma \geq 1$ in the spatial variables, we prove that the corresponding solution is Gevrey regular of order $5 \sigma$ in time. Moreover, we show that the function $u(x, y, t)$, viewed as a function of $t$, does not belong to $G^z$ for any $1 \leq z<5 \sigma$. The proof simultaneously treats all three variables $x, y$, and $t$, and employs the method of majorant series, precisely tracking the influence of the higher-order dispersive term $\partial_x^5 u$ together with the lower-order terms $\alpha \partial_x^3 u, \partial_x^{-1} \partial_y^2 u$, and $u \partial_x u$.