On the Gevrey regularity of the fifth-order Kadomtsev-Petviashvili-II equation: An improved approach

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-II 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 $\sigma \ge 1$ 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 $1\le z<5\sigma$. Our proof introduces a new analytical method that establishes a general principle for dispersive equations of the form ${\mathrm{\partial }}_{t}u=\pm {\mathrm{\partial }}_x^{\alpha }u+P(u),$ where ${\mathrm{\partial }}_x^{\alpha }$ is the highest spatial derivative and $P(u)$ a polynomial in spatial derivatives of total order at most $\alpha -1$, the solution cannot belong to the Gevrey class $G^z$ in time for any $z$ satisfying $1\le z<\alpha \sigma$.