On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems
Lastra and Malek Advances in Difference Equations
On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems
Alberto Lastra 1
Stéphane Malek 0
0 Laboratoire Paul Painlevé, University of Lille 1, Villeneuve d'Ascq Cedex , 59655 , France
1 Departamento de Física y Matemáticas, University of Alcalá, Ap. de Correos 20, Alcalá de Henares , Madrid E-28871 , Spain
We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter whose coefficients depend holomorphically on ( , t) near the origin in C2 and are bounded holomorphic on some horizontal strip in C w.r.t. the space variable. In our previous contribution (Lastra and Malek in Parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, arXiv:1403.2350), we assumed the forcing term of the Cauchy problem to be analytic near 0. Presently, we consider a family of forcing terms that are holomorphic on a common sector in time t and on sectors w.r.t. the parameter whose union form a covering of some neighborhood of 0 in C∗, which are asked to share a common formal power series asymptotic expansion of some Gevrey order as tends to 0. We construct a family of actual holomorphic solutions to our Cauchy problem defined on the sector in time and on the sectors in mentioned above. These solutions are achieved by means of a version of the so-called accelero-summation method in the time variable and by Fourier inverse transform in space. It appears that these functions share a common formal asymptotic expansion in the perturbation parameter. Furthermore, this formal series expansion can be written as a sum of two formal series with a corresponding decomposition for the actual solutions which possess two different asymptotic Gevrey orders, one stemming from the shape of the equation and the other originating from the forcing terms. The special case of multisummability in is also analyzed thoroughly. The proof leans on a version of the so-called Ramis-Sibuya theorem which entails two distinct Gevrey orders. Finally, we give an application to the study of parametric multi-level Gevrey solutions for some nonlinear initial value Cauchy problems with holomorphic coefficients and forcing term in ( , t) near 0 and bounded holomorphic on a strip in the complex space variable.
asymptotic expansion; Borel-Laplace transform; Fourier transform; Cauchy problem; formal power series; nonlinear integro-differential equation; nonlinear partial differential equation; singular perturbation
1 Introduction
We consider a family of parameter depending nonlinear initial value Cauchy problems of
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+ (δD–)(k+)–δD+t(δD–)(k+)∂tδD RD(∂z)udp (t, z, )
+ c(t, z, )R(∂z)udp (t, z, ) + cF ( )f dp (t, z, )
for given vanishing initial data udp (, z, ) ≡ , where D ≥ and δD, k, l, dl, δl, ≤ l ≤
D – are nonnegative integers and Q(X), Q(X), Q(X), Rl(X), ≤ l ≤ D, are
polynomials belonging to C[X]. The coefficient c(t, z, ) is a bounded holomorphic function on a
product D(, r) × Hβ × D(, ), where D(, r) (resp. D(, )) denotes a disc centered at
with small radius r > (resp. > ) and Hβ = {z ∈ C/| Im(z)| < β} is some strip of width
β > . The coefficients c,( ) and cF ( ) define bounded holomorphic functions on D(, )
vanishing at = . The forcing terms f dp (t, z, ), ≤ p ≤ ς – , form a family of bounded
holomorphic functions on products T × Hβ × Ep, where T is a small sector centered at
contained in D(, r) and {Ep}≤p≤ς– is a set of bounded sectors with aperture slightly
larger than π /k covering some neighborhood of in C∗. We make assumptions in order
that all the functions → f dp (t, z, ), seen as functions from Ep into the Banach space F of
bounded holomorphic functions on T × Hβ endowed with the supremum norm, share a
common asymptotic expansion fˆ(t, z, ) = m≥ fm(t, z) m/m! ∈ F of Gevrey order /k
on Ep, for some integer ≤ k < k; see Lemma .
Our main purpose is the construction of actual holomorphic solutions udp (t, z, ) to the
problem () on the domains T × Hβ × Ep and to analyze their asymptotic expansions as
tends to .
This work is a continuation of the study initiated in [] where the authors have studied
initial value problems with a quadratic nonlinearity of the form
Q(∂z) ∂tu(t, z, ) = Q(∂z)u(t, z, ) Q(∂z)u(t, z, )
+ (δD–)(k+)–δD+t(δD–)(k+)∂tδD RD(∂z)u(t, z, )
for given vanishing initial data u(, z, ) ≡ , where D, l, dl, δl are positive integers and
Q(X), Q(X), Q(X), Rl(X), ≤ l ≤ D, are polynomials with complex coefficients. Under (...truncated)