On the persistence of spatial analyticity for the Beam Equation

TL;DR

This paper investigates how the spatial analyticity of solutions to the beam equation persists over time, establishing a lower bound on the radius of analyticity that diminishes at a rate proportional to 1/√t as time progresses.

Contribution

It provides the first known asymptotic lower bound on the radius of spatial analyticity for solutions to the nonlinear beam equation over time.

Findings

The radius of analyticity decreases no faster than c/√t as t increases.

Solutions with initial analytic data maintain a quantifiable degree of analyticity over time.

The results apply to a class of initial data with uniform initial analyticity radius.

Abstract

Persistence of spatial analyticity is studied for solution of the beam equation $ u_{tt} + \left(m+\Delta^2\right) u + |u|^{p-1}u = 0$ on $\mathbb R^n \times \mathbb R$. In particular, for a class of analytic initial data with a uniform radius of analyticity $\sigma_0$, we obtain an asymptotic lower bound $\sigma(t) \ge c/\sqrt t$ on the uniform radius of analyticity $\sigma(t)$ of solution $u(\cdot, t)$, as $t \rightarrow \infty.$