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Gevrey regularity of the solutions of the inhomogeneous partial differential equations with a polynomial semilinearity

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Abstract

In this article, we are interested in the Gevrey properties of the formal power series solution in time of the partial differential equations with a polynomial semilinearity and with analytic coefficients at the origin of \({\mathbb {C}}^{n+1}\). We prove in particular that the inhomogeneity of the equation and the formal solution are together s-Gevrey for any \(s\ge s_c\), where \(s_c\) is a nonnegative rational number fully determined by the Newton polygon of the associated linear PDE. In the opposite case \(s<s_c\), we show that the solution is generically \(s_c\)-Gevrey while the inhomogeneity is s-Gevrey, and we give an explicit example in which the solution is \(s'\)-Gevrey for no \(s'<s_c\).

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Notes

  1. We denote \(\widetilde{f}\) with a tilde to emphasize the possible divergence of the series \(\widetilde{f}\).

  2. We have thereby \(s_c=1\) in the case of Eq. (1.3); and, in the case of Eq. (1.4), \(s_c=p/\kappa -1\) when \(p>\kappa \), and \(s_c=0\) when \(p\le \kappa \). For the definition of the Newton polygon, we refer to Sect. 2 below.

  3. This is obvious due to the filtration of the Gevrey spaces (see Sect. 3.1) and the first point of Proposition 1.

  4. These numbers were named in honor of the mathematician Eugène Charles Catalan (1814–1894). They appear in many probabilist, graphs and combinatorial problems. For example, they can be seen as the number of d-ary trees with m source-nodes, or as the number of ways of associating m applications of a given d-ary operation, or as the number of ways of subdividing a convex polygon into m disjoint (\(d+1\))-gons by means of non-intersecting diagonals. They also appear in theoretical computers through the generalized Dyck words. See for instance [4] and the references inside.

  5. Of course, this case only occurs when \({\mathcal {S}}\ne \emptyset \).

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  1. Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin, 45 Avenue des Etats-Unis, 78035, Versailles Cedex, France

    Pascal Remy

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Remy, P. Gevrey regularity of the solutions of the inhomogeneous partial differential equations with a polynomial semilinearity. RACSAM 115, 145 (2021). https://doi.org/10.1007/s13398-021-01085-5

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