On the Domain of Analyticity for Solutions of Second Order Analytic Nonlinear Differential Equations

The radius of analyticity of periodic analytic functions can be characterized by the decay of their Fourier coefficients. This observation has led to the use of so-called Gevrey norms as a simple way of estimating the time evolution of the spatial radius of analyticity of solutions to parabolic as well as non-parabolic partial differential equations. In this paper we demonstrate, using a simple, explicitly solvable model equation, that estimates on the radius of analyticity obtained by the usual Gevrey class approach do not scale optimally across a family of solutions, nor do they scale optimally as a function of the physical parameters of the equation. We attribute the observed lack of sharpness to a specific embedding inequality, and give a modified definition of the Gevrey norms which is shown to finally yield a sharp estimate on the radius of analyticity.