On the summability of formal solutions for doubly singular nonlinear partial differential equations

Abstract

We study Gevrey asymptotic properties of solutions of singularly perturbed singular nonlinear partial differential equations of irregular type in the complex domain. We construct actual holomorphic solutions of these problems with the help of the BorelLaplace transforms. Using the Malgrange-Sibuya theorem, we show that these holomorphic solutions have a common formal power series asymptotic expansion of Gevrey order 1 in the perturbation parameter.

On singularly perturbed small step size difference-differential nonlinear PDEs

“…In a first part, we construct a holomorphic function V (τ, z, ) near the origin with respect to (τ, z) and on a punctured disc with respect to which solves an integro-differential problem whose coefficients are meromorphic functions with respect to (τ, ) with a pole at = 0, see (108), (109). The main novelty compared to our previous studies on singular perturbation problems, see [21], [23], [24], [25], is that the coefficients of (108) now have at most polynomial growth with respect to τ on the half plane C + = {τ ∈ C/Re(τ ) ≥ 0} but exponential growth on the half plane C − = {τ ∈ C/Re(τ ) < 0}. For suitable initial data satisfying the conditions (117), (118), we show that V (τ, z, ) can be analytically continued to functions…”

“…In the proof, we use as in [24] deformations of the integration's paths in X i with the help of the estimates (7) and (8) (Theorem 1).…”

“…In the paper [24], we have considered singular singularly perturbed nonlinear problems (3) t 2 ∂ t ∂ S z u i (t, z, ) = F (t, z, , ∂ t , ∂ z )u i (t, z, ) + P (t, z, , u i (t, z, )) for given initial data (4) (∂ j z u i )(t, 0, ) = φ j,i (t, ) , 0 ≤ i ≤ ν − 1, 0 ≤ j ≤ S − 1, where F is some differential operator with polynomial coefficients and P some polynomial. The initial data φ j,i (t, ) were assumed to be holomorphic on products (T ∩ {|t| < h }) × E i , for some h > 0 small enough.…”

“…In this work we address the same question as in our previous papers [21], [24], namely our main goal is the construction of actual holomorphic solutions X i (t, z, ) to the problem (1), (2) on domains (T ∩ {|t| > h}) × D(0, δ) × E i for some small disc D(0, δ) and the analysis of their asymptotic expansions as tends to 0. More precisely, we can present our main statements as follows.…”

Parametric Borel summability for some semilinear system of partial differential equations

“…On the other hand, concerning the summability of formal solutions of a partial differential equation with a singular perturbation parameter we cite [2] and [4]. (See also [6,7] and [9]. )…”