Kurogi, Kenji Counterexample to the 1-summability of a divergent formal solution to some linear partial differential equations. (English) Zbl 1426.35071 Funkc. Ekvacioj, Ser. Int. 61, No. 2, 219-227 (2018). The work under review provides a counterexample to the Borel-Laplace summability property of the formal solutions to certain family of linear partial differential equations of the form \[ \partial_t^m u+\sum_{(j,\alpha)\in\Lambda}a_{j,\alpha}(t)\partial_t^j\partial_x^{\alpha}u=f(t,x) \] under initial conditions \(\partial_{t}^{j}u|_{t=0}=\varphi_j(x)\), for \(j=0,\ldots,m-1\), where \(\Lambda\) is a finite subset of \(\mathbb{N}\times \mathbb{N}^{N}\), the coefficients \(a_{j,\alpha}(t)\) are holomorphic functions on some neighborhood of the origin, \(f\) is holomorphic on some neighborhood of the origin with respect to its first variables and an entire function with respect to \(x\), and the initial data are entire functions.The author gives an example in which one of the conditions imposed in the previous work by H. Tahara and H. Yamazawa [J. Differ. Equations 255, No. 10, 3592–3637 (2013; Zbl 1320.35151)] on the 1-summability of the formal solution of the previous problem does not hold. In that example, the loss of such condition shows that the formal solution is not 1-summable along any direction. Reviewer: Alberto Lastra Sedano (Burgos) MSC: 35C10 Series solutions to PDEs 35A20 Analyticity in context of PDEs Keywords:partial differential equations; summability; Newton polygon Citations:Zbl 1320.35151 PDFBibTeX XMLCite \textit{K. Kurogi}, Funkc. Ekvacioj, Ser. Int. 61, No. 2, 219--227 (2018; Zbl 1426.35071) Full Text: DOI References: [1] Balser, W., Formal power series and linear systems of meromorphic ordinary differential equations, Springer-Verlag, New York, 2000. · Zbl 0942.34004 [2] Hibino, M., Borel summability of divergent solutions for singular first order linear partial differential equations with polynomial coefficients, J. Math. Sci. Univ. Tokyo, 10 (2003), 279-309. · Zbl 1036.35051 [3] Lutz, D. A., Miyake, M. and Schäfke, R., On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J., 154 (1999), 1-29. · Zbl 0958.35061 [4] Malek, S., On the summability of formal solutions of linear partial differential equations, J. Dyn. Control Syst., 11 (2005), 389-403. · Zbl 1085.35043 [5] Ōuchi, S., Genuine solutions and formal solutions with Gevrey type estimates of nonlinear partial differential equations, J. Math. Sci. Univ. Tokyo, 2 (1995), 375-417. · Zbl 0860.35018 [6] Ōuchi, S., Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations, 185 (2002), 513-549. · Zbl 1020.35018 [7] Ōuchi, S., Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains, Asymptot. Anal., 47 (2006), 187-225. · Zbl 1152.35015 [8] Tahara, H. and Yamazawa, H., Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations, J. Differential Equations, 255 (2013), 3592-3637. · Zbl 1320.35151 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.