Opuscula Math. 35, no. 5 (2015), 825-845
Parametric Borel summability for some semilinear system of partial differential equations
Abstract. In this paper we study the Borel summability of formal solutions with a parameter of first order semilinear system of partial differential equations with \(n\) independent variables. In [Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002), 313-322], Balser and Kostov proved the Borel summability of formal solutions with respect to a singular perturbation parameter for a linear equation with one independent variable. We shall extend their results to a semilinear system of equations with general independent variables.
Keywords: Borel summability, singular perturbation, Euler type operator.
Mathematics Subject Classification: 35C10, 45E10, 35Q15.
- W. Balser, Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, Springer-Verlag, New York (2000).
- W. Balser, V. Kostov, Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002) 3, 313-322.
- W. Balser, M. Loday-Richaud, Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables, Adv. Dynam. Syst. Appl. 4 (2009), 159-177.
- W. Balser, J. Mozo-Fernández, Multisummability of formal solutions of singular perturbation problems, J. Differential Equations 183 (2002), 526-545.
- K. Ichinobe, Integral representation for Borel sum of divergent solution to a certain non-Kowalevski type equation, Publ. Res. Inst. Math. Sci. 39 (2003), 657-693.
- Z. Luo, H. Chen, C. Zhang, Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations, Ann. Inst. Fourier 62 (2012) 2, 571-618.
- A. Lastra, S. Malek, J. Sanz, On Gevrey solutions of threefold singular nonlinear partial differential equations, J. Differential Equations 255 (2013) 10, to appear.
- D.A. Lutz, M. Miyake, R. Schäfke, On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J. 154 (1999), 1-29.
- S. Malek, On the summability of formal solutions for doubly singular nonlinear partial differential equations, J. Dynam. Control. Syst. 18 (2012), 45-82.
- S. Michalik, Summability of formal solutions to the \(n\)-dimensional inhomogeneous heat equation, J. Math. Anal. Appl. 347 (2008), 323-332.
- S. Ouchi, Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains, Asympt. Anal. 47 (2006) 3-4, 187-225.
- Communicated by Theodore A. Burton.
- Received: 2013-11-26.
- Revised: 2015-03-19.
- Accepted: 2015-03-20.
- Published online: 2015-04-27.