On Parametric Gevrey Asymptotics for Some Initial Value Problems in Two Asymmetric Complex Time Variables

Abstract

We study a family of nonlinear initial value partial differential equations in the complex domain under the action of two asymmetric time variables. Different Gevrey bounds and multisummability results are obtain depending on each element of the family, providing a more complete picture on the asymptotic behavior of the solutions of PDEs in the complex domain in several complex variables.The main results lean on a fixed point argument in certain Banach space in the Borel plane, together with a Borel summability procedure and the action of different Ramis-Sibuya type theorems. variable coefficients (see [5,6,21]). The second author [22], both authors [13] and the two authors and J. Sanz [17] have also contributed in this theory. Recently, multisummability of formal solutions of PDEs has also been put forward in different works. W. Balser [4] described a multisummability phenomenon in certain PDEs with constant coefficients. S. Ouchi [23] constructed multisummable formal solutions of nonlinear PDEs, coming from perturbation of ordinary differential equations. H. Tahara and H. Yamazawa [24] have made progresses on general linear PDEs with non constant coefficients under entire initial data. In [20], G. Lysik constructs summable formal solutions of the one dimensional Burgers equation by means of the Cole-Hopf transform. O. Costin and S. Tanveer [8] construct summable formal power series in time variable to 3D Navier Stokes equations. The authors have obtained results in this direction [14, 15].A recent overview on summability and multisummability techniques under different points of view is displayed in [18].The purpose of the present work is to study the solutions of a family of singularly perturbed partial differential equations from the asymptotic point of view. More precisely, we consider a problem of the formThe elements which conform the nonlinear part P 1 , P 2 are polynomials in their second variable with coefficients being holomorphic functions defined on some neighborhood of the origin, say D(0, 0 ), continuous up to their boundary.Here, D(0, 0 ) stands for the open disc in the complex plane centered at 0, and with positive radius 0 > 0. We write D(0, 0 ) for its closure. Moreover, P stands for some polynomial of six variables, with complex coefficients, and the forcing term f (t 1 , t 2 , z, ) is a holomorphic and bounded function in D(0, ρ) 2 × H β × D(0, 0 ), for some ρ > 0, and where H β stands for the horizontal stripfor some β > 0.The precise configuration of the elements involved in the problem is stated and described in Section 2.2.This paper provides a step beyond in the study of the asymptotic behavior of the solutions of a subfamily of singularly perturbed partial differential equations of the form (1). We first recall some previous advances made in this respect, which motivate the present framework.In [13], we studied under the asymptotic point of view the solutions of certain family of PDEs of the formwhere the elements involved in the problem depend only on one time variable t. Our next aim w...

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

“…In view of the failure of the approaches in [13,14] (see Section 3.1), we need to adopt another perspective.…”

“…along well chosen halflines L d j ⊂ S j , j = 1, 2, as it was the case in our previous studies [13,14]. Our idea consists on merging this double integral along the product L d 1 × L d 2 into a simple integral along a halfline L d ⊂ C. Geometrically, it consists in a projection on the diagonal part (u, u) ∈ C 2 , u ∈ C of the space C 2 .…”

“…The Banach spaces described in this section are modified versions of those appearing in [15]. We omit the details which can be derived directly from that work, and the variations of that norm, stated in [13,14].…”

“…In the second study [14], the property of symmetry of the equations drops, and P in (3) is no longer factorizable into two terms which only present dependence on one of the time variables. This asymmetry causes that the procedure followed in [13] is no longer valid in that second framework and the procedure followed differs from that in [13].…”

On a q—Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables

“…Let d ∈ R be such that the infinite sector S d of bisecting direction d satisfies the following geometric construction: there exists D(0, ρ). The previous is a feasible condition for an appropriate choice of small enough η Q,R D 1 D 2 > 0 and large enough r Q,R D 1 D 2 > 0, in view of (14) and the definition of q (m). More precisely, one chooses S d such that q (m)/τ has positive distance to 1, for all 0…”