Parametric Borel summability for some semilinear system of partial differential equations

Abstract

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.

Boundary layer expansions for initial value problems with two complex time variables

“…The study of singularly perturbed PDEs in the complex domain is a topic of increasing interest. In 2015, H. Yamazawa and M. Yoshino [23] studied parametric Borel summability in semilinear systems of PDEs of fuchsian type, and of combined irregular and fuchsian type by M. Yoshino [24].…”

Analytic continuation of Borel sum of formal solution of semilinear partial differential equation

“…Clearly, the set π/2 < arg ξ < 3π/2 is equal to S π,π , which is the set corresponding to the case where C 0 is a positive real line. As for the proof of this statement we refer to [11,Theorem 3]. If V (x, η) is the 1-Borel sum of v(x, η) defined in some neighborhood of a and η ∈ S 2π,π , then V (x, η) is characterized as the solution of (2.2) having an asymptotic expansion with respect to η when η ∈ S θ,π , η → ∞ and x is in some neighborhood of a for some θ > π.…”

“…On the other hand, the summability of partial differential equations with a singular perturbative parameter was studied by Malek et al, where asymptotic solutions were constructed (cf. [5,7] and [11]). In general, the summability breaks down at a singular direction.…”

On parametric Gevrey asymptotics for some nonlinear initial value problems in symmetric complex time variables

“…In recent years, an increasing interest on complex singularly perturbed PDEs has been observed in the area. Parametric Borel summability has been described in semilinear systems of PDEs of Fuchsian type by H. Yamazawa and M. Yoshino in [16] η n j=1…”