On the persistence of spatial analyticity for generalized KdV equation with higher order dispersion

Tegegne Getachew, Achenef Tesfahun and Birilew Belayneh

Mathematische Nachrichten, 2024, vol. 297, issue 5, 1737-1748

Abstract: Persistence of spatial analyticity is studied for solutions of the generalized Korteweg‐de Vries (KdV) equation with higher order dispersion ∂tu+(−1)j+1∂x2j+1u=∂xu2k+1,$$\begin{equation*} \partial _{t} u+(-1)^{j+1}\partial _{x}^{2j+1} u= \partial _x{\left(u^{2k+1} \right)}, \end{equation*}$$where j≥2$j\ge 2$, k≥1$k\ge 1$ are integers. For a class of analytic initial data with a fixed radius of analyticity σ0$\sigma _0$, we show that the uniform radius of spatial analyticity σ(t)$\sigma (t)$ of solutions at time t$t$ cannot decay faster than 1t$\frac{1}{\sqrt t}$ as t→∞$t\rightarrow \infty$. In particular, this improves a recent result due to Petronilho and Silva [Math. Nachr. 292 (2019), no. 9, 2032–2047] for the modified Kawahara equation (j=2$j=2$, k=1$k=1$), where they obtained a decay rate of order t−4+$ t^{-4 +}$. Our proof relies on an approximate conservation law in a modified Gevrey spaces, local smoothing, and maximal function estimates.

Date: 2024
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