Improved lower bound of spatial analyticity radius for solutions to nonlinear wave equation

Document Type : Research Article

Authors

Department of Mathematics, Mekdela Amba University, Ethiopia

Abstract

In this paper, the rate of decay for the radius of spatial analyticity for solutions of the nonlinear wave equation
\[\partial_t^2 u -\Delta u + |u|^{p-1}u=0, \]
on $\mathbb{R}^d\times\mathbb{R}$ is studied. In particular, for a
class of analytic initial data with a uniform radius of analyticity $\sigma_0$, we obtain an asymptotic lower bound $\sigma(t)\ge a_0|t|^{-\frac23}$ when $d=1$ and $\sigma(t)\ge a_0|t|^{-\frac32}$ when $d=2$
on the uniform radius of analyticity $\sigma(t)$ of solution $u(\cdot,t)$ as $|t|\rightarrow +\infty$ . This is an improvement of the work [D.~O.~da~Silva, A.~J.~Castro, Global well-posedness for the nonlinear wave equation in analytic Gevrey spaces, J. Differential Equations 275(2021)~234--249], where the authors obtained a decay rate of order $\sigma(t)\geq a_0(1+|t|)^{-(\frac{p+1}{2})}$ when $d=1$ and $\sigma(t)\geq a_0(1+|t|)^{-(\frac{p+1-\epsilon}{1-\epsilon})}$ when $d=2$ as $|t|\rightarrow +\infty$ for large time $t$, where $\epsilon>0$ is arbitrary. We used an approximate conservation law in a modified Gevrey space, contraction mapping principle, interpolation and Sobolev embedding to obtain the results.

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Journal of Mathematical Modeling
Volume 14, Issue 1
March 2026
Pages 347-361

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