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

10.22124/jmm.2025.30485.2735

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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