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Gevrey regularity of the solutions of some inhomogeneous semilinear partial differential equations with variable coefficients

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Abstract

In this article, we are interested in the Gevrey properties of the formal power series solution in time of some partial differential equations with a power-law nonlinearity and with analytic coefficients at the origin of \({\mathbb {C}}^2\). We prove in particular that the inhomogeneity of the equation and the formal solution are together s-Gevrey for any \(s\geqslant s_c\), 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. Observe that this fact is well-known in the case of the ODEs: the Gevrey order of the formal solutions of any semilinear meromorphic ordinary differential equations is given by the Newton polygon of its linear part—see [7, 47] for instance.

  3. The case \(\beta (X)\equiv 0\) corresponds to the linear case, and the case \(h(X)\equiv 0\) corresponds to the homogeneous case with the initial condition \(\varphi _j(x)\equiv 0\) for all \(j=0,...,\kappa -1\).

  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 m-ary trees with i source-nodes, or as the number of ways of associating i applications of a given m-ary operation, or as the number of ways of subdividing a convex polygon into i disjoint (\(m+1\))-gons by means of non-intersecting diagonals. They also appear in theoretical computers through the generalized Dyck words. See for instance [13] and the references inside.

  5. Of course, this case only occurs when \(p>\kappa \).

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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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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.

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Remy, P. Gevrey regularity of the solutions of some inhomogeneous semilinear partial differential equations with variable coefficients. Partial Differ. Equ. Appl. 4, 19 (2023). https://doi.org/10.1007/s42985-023-00236-0

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