Electron. J. Differential Equations, Vol. 2023 (2023), No. 06, pp. 128.
Gevrey regularity of the solutions of inhomogeneous nonlinear partial differential equations
Pascal Remy
Abstract:
In this article, we are interested in the Gevrey properties of the formal power series
solutions in time of some inhomogeneous nonlinear partial differential equations with
analytic coefficients at the origin of C^{n+1}. We systematically examine the cases
where the inhomogeneity is sGevrey for any s≥0, in order to carefully distinguish
the influence of the data (and their degree of regularity) from that of the equation
(and its structure). We thus prove that we have a noteworthy dichotomy with respect to
a nonnegative rational number s_{c} fully determined by the Newton polygon of a
convenient associated linear partial differential equation: for any s≥s_{c},
the formal solutions and the inhomogeneity are simultaneously sGevrey; for any
s<s_{c}, the formal solutions are generically s_{c}Gevrey.
In the latter case, we give an explicit example in which the solution is s'Gevrey
for no s'<s_{c}. As a practical illustration, we apply our results to the generalized
BurgersKortewegde Vries equation.
Submitted November 6, 2021. Published January 19, 2023.
Math Subject Classifications: 35C10, 35G20, 35Q53.
Key Words: Gevrey order; inhomogeneous partial differential equation;
nonlinear partial differential equation; generalized BurgersKdV equation;
Newton polygon; formal power series; divergent power series.
DOI: https://doi.org/10.58997/ejde.2023.06
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Pascal Remy
Laboratoire de Mathématiques de Versailles
Université de Versailles SaintQuentin
45 avenue des EtatsUnis
78035 Versailles cedex, France
email: pascal.remy@uvsq.fr; pascal.remy.maths@gmail.com

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