Abstract
We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter ϵ whose coefficients depend holomorphically on $(\epsilon,t)$ near the origin in $\mathbb{C}^{2}$ and are bounded holomorphic on some horizontal strip in $\mathbb{C}$ 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 ϵ whose union form a covering of some neighborhood of 0 in $\mathbb{C}^{\ast}$ , which are asked to share a common formal power series asymptotic expansion of some Gevrey order as ϵ 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 ϵ 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 ϵ 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 $(\epsilon,t)$ near 0 and bounded holomorphic on a strip in the complex space variable.
Highlights
We consider a family of parameter depending nonlinear initial value Cauchy problems of the formQ(∂z) ∂tudp (t, z, ) = c, ( ) Q (∂z)udp (t, z, ) Q (∂z)udp (t, z, )Lastra and Malek Advances in Difference Equations (2015) 2015:200+(k + )–δD+ t(δD– )(k + )∂tδD RD(∂z)udp (t, z, ) D– + l=l tdl ∂tδl Rl(∂z)udp (t, z, )+ c (t, z, )R (∂z)udp (t, z, ) + cF ( )f dp (t, z, ) ( )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
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
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
Summary
We construct a formal power series U (T, m, ) = n≥ Un(m, )Tn solution of ( ) whose coefficients m → Un(m, ) depend holomorphically on near and belong to a Banach space E(β,μ) of continuous functions with exponential decay on R introduced by Costin and Tanveer in [ ]. Under some size constraints on the sup norm of the coefficients c , ( )/ , c ( )/ and cF ( )/ near , we show that ωk (τ , m, ) is convergent for τ on some fixed neighborhood of and can be extended to a holomorphic function ωkd (τ , m, ) on unbounded sectors Ud centered at with bisecting direction d and tiny aperture, provided that the mk -Borel transform of the formal forcing term F(T, m, ), denoted by ψk (τ , m, ) is convergent near τ = and can be extended on Ud w.r.t. τ as a holomorphic function ψkd (τ , m, ) with exponential growth of order less than k .
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