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

In this paper, the rate of decay for the radius of spatial analyticity for solutions of the nonlinear wave equation \[\partial_t^۲ u -\Delta u + |u|^{p-۱}u=۰, \] on \mathbb{R}^d\times\mathbb{R} is studied. In particular, for aclass of analytic initial data with a uniform radius of analyticity \sigma_۰, we obtain an asymptotic lower bound \sigma(t)\ge a_۰|t|^{-\frac۲۳} when d=۱ and \sigma(t)\ge a_۰|t|^{-\frac۳۲} when d=۲ 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 ۲۷۵(۲۰۲۱)~۲۳۴--۲۴۹], where the authors obtained a decay rate of order \sigma(t)\geq a_۰(۱+|t|)^{-(\frac{p+۱}{۲})} when d=۱ and \sigma(t)\geq a_۰(۱+|t|)^{-(\frac{p+۱-\epsilon}{۱-\epsilon})} when d=۲ as |t|\rightarrow +\infty for large time t, where \epsilon>۰ is arbitrary. We used an approximate conservation law in a modified Gevrey space, contraction mapping principle, interpolation and Sobolev embedding to obtain the results.

کلیدواژه ها:

Nonliear wave equation ، Modified Gevrey space ، Approximate conservation ، Radius of analyticity ، Decay rate for the radius