M. Oliver et Es. Titi, On the domain of analyticity for solutions of second order analytic nonlinear differential equations, J DIFF EQUA, 174(1), 2001, pp. 55-74
The radius of analyticity of periodic analytic functions can be characteriz
ed 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 ev
olution of the spatial radius of analyticity of solutions to parabolic as w
ell as non-parabolic partial differential equations. In this paper we demon
strate, 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, not do they scale optima
lly 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. 2001 Academic Press.