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M. Oliver and E.S. Titi,
On the domain of analyticity for solutions of second
order analytic nonlinear differential equations,
J. Differential Equations 174 (2001), 55-74.
Abstract:
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.
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