We examine a linear q−difference differential equation, which is singular in complex time t at the origin. Its coefficients are polynomial in time and bounded holomorphic on horizontal strips in one complex space variable. The equation under study represents a q−analog of a singular partial differential equation, recently investigated by the author, which comprises Fuchsian operators and entails a forcing term that combines polynomial and logarithmic type functions in time. A sectorial holomorphic solution to the equation is constructed as a double complete Laplace transform in both time t and its complex logarithm logt and Fourier inverse integral in space. For a particular choice of the forcing term, this solution turns out to solve some specific nonlinear q−difference differential equation with polynomial coefficients in some positive rational power of t. Asymptotic expansions of the solution relatively to time t are investigated. A Gevrey-type expansion is exhibited in a logarithmic scale. Furthermore, a formal asymptotic expansion in power scale is displayed, revealing a new fine structure involving remainders with both Gevrey and q−Gevrey type growth.

1. Introduction

In this work, we draw attention to a family of singular linear q−difference differential equations modeled as

()

for vanishing initial data u(0, z) ≡ 0, where d_D, k₁ ≥ 1 from the leading term of Equation (1) are integers, Q(X), R_D(X) represent polynomials with complex coefficients and P(t, z, V₁, V₂) stands for a polynomial in its arguments t, V₁, V₂ with holomorphic coefficients relatively to the space variable z on a horizontal strip in

designed as

, for some prescribed width 2β > 0. The forcing term f(t, z) is a logarithmic type function represented as a polynomial in both complex time variable t and inverse of its complex logarithm 1/logt with coefficients that are bounded holomorphic on the strip H_β.

This paper is a natural sequel of the recent study of Malek [1]. Indeed, in [1], we focused on the next singularly perturbed linear partial differential equations shaped as

()

for given initial data y(0, z, ɛ) ≡ 0, for integers d_D, k₁ ≥ 1 appearing in the principal term of Equation (2), complex polynomials Q(X), R_D(X) as above and where H(t, z, ɛ, V₁, V₂) represents a polynomial in t, V₁, V₂ whose coefficients are bounded holomorphic w.r.t z on the strip H_β and relatively to a complex parameter ɛ on some fixed disc

centered at 0 for some radius ɛ₀ > 0. The forcing term h(t, z, ɛ) comprises coefficients that rely polynomially on complex time t, analytically in ɛ on

and holomorphically in z on H_β. This term also entails logarithmic type functions stated as truncated Laplace transforms along a fixed segment [−a, 0] for some radius a > 0 that involve the inverse complex logarithm 1/log(ɛt). When the radius a > 0 is taken large, the expression of the forcing term h becomes proximate to maps that are similar to the forcing term f(t, z) of Equation (1) described above, namely a polynomial in both ɛt and 1/log(ɛt) with bounded holomorphic coefficients on

Under suitable constraints set on the profile of Equation (2), we were able to construct a set of genuine bounded holomorphic solutions y_p to Equation (2), for p in a finite subset I₁ of the natural numbers

, expressed as a complete Laplace transform of integer order k₁ in the monomial ɛt, a truncated Laplace transform of order 1 in the inverse 1/log(ɛt) and inverse Fourier integral in the space variable z,

()

where the so-called Borel–Fourier map w_p(τ₁, τ₂, m, ɛ) is as follows:

(1)
Analytic near τ₁ = 0 and relatively to τ₂ ∈ D_a and and has (at most) exponential growth of order k₁ along some well-chosen unbounded sector centered at 0 and containing the halfline for , with respect to τ₁.
(2)
Continuous and subjected to exponential decay in phase .

As a result, these functions y_p(t, z, ɛ) define bounded holomorphic maps on domains

, for well selected bounded sector

edged at 0 and where

is an appropriate set of bounded sectors centered at 0. At this point, it is crucial to notice that these solutions y_p cannot be represented as complete Laplace transform in the map 1/log(ɛt). It turns out that the radii a, ɛ₀ > 0 are related by a rule of the form

, for some suitable constant M > 0 and positive integers n₀, n₁ ≥ 1.

Besides, asymptotic features of these solutions have been examined in [1]. It appears that the family

owns asymptotic expansions of Gevrey type in two distinguished scales of functions. Indeed, for each p ∈ I₁, the partial map (t, ɛ) ↦ y_p(t, z, ɛ) holds a generalized asymptotic formal expansion (in a sense defined in the classical textbooks [2, 3]).

()

on the domain

, in the scale of logarithmic functions

with bounded holomorphic coefficients

. These asymptotic expansions are revealed to be of Gevrey 1 on

, giving rise to constants K¹, M¹ > 0 for which the error bounds

()

occur for all integers N ≥ 0, provided that

, where Γ(x) stands for the Gamma function in x. On the other hand, all the partial maps ɛ ↦ y_p(t, z, ɛ), p ∈ I₁, share a common generalized asymptotic formal expansion

()

, in the scale of monomial

with bounded holomorphic coefficients

, for some open domain

containing all the sectors

, p ∈ I₁. Moreover, these asymptotic expansions happen to be of Gevrey order 1/k₁ on each sector, meaning that constants

can be pinpointed for which the error estimates

()

hold for all integers N ≥ 0, whenever

. At last, we proved in [1] that the coefficients

and

of both formal expansions

and

solve explicit differential recursion relations with respect to n ≥ 0 that might be handy for effective computations.

In the present investigation of the problem (1), we plan to follow a similar roadmap as in [1]. Namely, we plan to build up genuine sectorial solutions to Equation (1) and describe their asymptotic expansions as time t borders the origin, instead of a perturbation parameter ɛ which does not appear in Equation (1). We notice that our main problem (1) can be viewed as a q−analog of Equation (2) where the Fuchsian operator t∂_t is substituted by the discrete dilation operator σ_q;t. This terminology stems from the plain observation that the quotient (f(qt) − f(t))/(qt − t) neighbors the derivative f^′(t) as q tends to 1. Problem (2) involves at first sight only powers of the basic differential operator of Fuchsian type t∂_t. However, the conditions imposed on Equation (2) allow to express it also by means of powers of the basic differential operator of so-called irregular type . The same fact is acknowledged for the problem (1) under study for which q−difference operators of the form where l₀ ≥ k₁l₁ appear, see Equation (30). These operators are labeled of irregular types in the literature by analogy with the differential case. We quote the classical textbooks [4, 5] as references for analytic aspects of differential equations with irregular type and the book [6] for analytic and algebraic features of q−difference equations with irregular type. This suggests that in the building process of the solutions to Equation (1), the classical Laplace transform of order k₁ ought to be supplanted by a q−Laplace transform of order k₁.

We now describe a little more precisely the main statements of this paper achieved in Theorems 1 and 2. Namely, under fitting restrictions on the shape of Equation (1) listed in Section 2.2 and complemented in the statement of Theorem 1 in Section 4.3, we can establish the existence of a bounded holomorphic solution u(t, z) to Equation (1) on a domain

, for some small radius R₁ > 0, where

stands for an open sector centered at 0 with large opening that does not contain the halfline

, see Equation (22), for thoroughly chosen directions

. In addition, the map u(t, z) is modeled through a triple integral, which entails a Fourier inverse, a q−Laplace, and a complete Laplace transforms

()

where the Borel–Fourier map

is as follows:

(1)
Analytic on a unbounded sector centered at 0 containing the halfline with respect to τ₁ where it has (at most) q−exponential growth of order k₁.
(2)
Analytic relatively to τ₂ on some open halfstrip.
()
with small width η₂ > 0 and on a small disc D_ρ.
(3)
Continuous and submitted to exponential decay in phase .

At this stage, we emphasize that the geometry of the Borel space in the variable (τ₁, τ₂) for the map differs significantly from the one of the Borel–Fourier map w_p in Equation (3). Indeed, the map is in general not analytic near τ₁ = 0 while w_p(τ₁, τ₂, m, ɛ) possesses this property. As we will realize later on, this will be the root of the dissemblances observed between the asymptotic properties of the solutions y_p of Equation (2) and the solution u of Equation (1). Besides, the partial map τ₂ ↦ w_p(τ₁, τ₂, m, ɛ) is only holomorphic on some fixed disc D_a but is analytic on a full halfstrip H_π which allows the solution u(t, z) to be expressed as a complete Laplace transform in 1/logt in direction π while y_p(t, z, ɛ) is represented as a truncated Laplace transform along the segment [−a, 0]. A direct byproduct of this observation is that the forcing term f(t, z) of Equation (1) can be presented as an exact polynomial in both time t and inverse complex logarithm 1/logt while the forcing term h(t, z, ɛ) has to be only considered as proximate to such a polynomial in t and 1/logt. Some interesting aftermath is reached when f(t, z) is chosen a mere monomial in t and 1/logt since in that case f(t, z) solves an explicit nonlinear ordinary differential equation with polynomial coefficients in some positive rational power t^α, , displayed in Equation (48). As a result, u(t, z) turns out to be an exact holomorphic solution to some specific nonlinear q−difference differential equation with bounded holomorphic coefficients with respect to z on H_β and polynomial in t^α, stated in Equation (50). Contrastingly, Equation (2) becomes close to some nonlinear partial differential equation as a → +∞, but no information can be extracted about the existence of an exact genuine solution to the limit nonlinear problem.

It is worthwhile noting that in recent years, much attention has been drawn on nonlinear q−difference equations and especially on those related to the so-called q−Painlevé equations. For a comprehensive overview on major studies for q−Painlevé equations and, more generally, for integrable discrete dynamical systems, we refer to the book [7]. In this trend of research we quote the novel paper [8], where the authors construct convergent generalized power series with complex exponents on sectors that are solutions to nonlinear algebraic q−difference equations. In the context of nonlinear q−difference differential equations, we mention an important result by Yamazawa [9]. Indeed, he considers equations with the shape

()

for

, for some integers n, m ≥ 1, some real number q > 1, where F is a well prepared analytic function in its arguments. Under nonresonance conditions of the so-called characteristic exponent ρ(x) associated to Equation (10) at x = 0, he has constructed convergent logarithmic type solutions of the form

()

where the coefficients u_i(x) and φ_i,j,k(x) are holomorphic on a common disc D_R and where ρ_q(x) = log(1 + (q − 1)ρ(x))/log(q) stands for a q−analog of the characteristic exponent ρ(x).

In the second part of Theorem 1, we exhibit for the solution u(t, z) of Equation (1) a generalized asymptotic expansion of Gevrey type in a logarithmic scale for t in the vicinity of 0. The statement is similar to the one reached in [1] for the solutions y_p(t, z, ɛ) of Equation (2). Indeed, the partial map t ↦ u(t, z) is shown to possess a generalized formal series

()

with bounded holomorphic coefficients G_n(t, z) on the domain

as asymptotic expansion of Gevrey order 1 with respect to t on

, leading to estimates of the form

()

for some constants K, M > 0, for all integers N ≥ 0, whenever

. Furthermore, in Section 4.4, Proposition 6, we provide explicit and simple q−difference and differential recursion relations displayed in Equations (189) and (190) for the coefficients G_n(t, z), n ≥ 0, intended for practical use. The existence of such a formal expression (12) is shown in a comparable way as Equation (4) for the partial maps (t, ɛ) ↦ y_p(t, z, ɛ) in the problem (2). Namely, it is based on sharp estimates of some exponential decay for the differences of neighboring analytic solutions

, disclosed in Equation (143), to some related q−difference differential equation which comprises an homography action, see Equations (139) and (141) in Proposition 4. In the process, we use a classical result known as the Ramis-Sibuya theorem (see Theorem (R.S.) in Section 4.2), which ensures the existence of a common Gevrey asymptotic expansion for families of sectorial holomorphic functions.

In the second main result of this paper, stated in Theorem 2, a generalized the asymptotic expansion of the solution u(t, z) is established in the scale of monomials

. This statement differs notably from the one obtained for the partial maps ɛ ↦ y_p(t, z, ɛ) in the problem (2). Namely, the holomorphic solution u(t, z) to Equation (1) can be split into a sum u(t, z) = u₁(t, z) + u₂(t, z) where

(1)
the map u₁(t, z) owns a formal expression
()
with bounded holomorphic coefficients b_n(t, z) on the domain as generalized asymptotic expansion of so-called q−Gevrey order k₁. It means that two constants B₁, B₂ > 0 can be found with the error bounds
()
for all integers N ≥ 0, all .
(2)
The map u₂(t, z) has the null formal series as asymptotic expansion of order 1 in a logarithmic scale as t tends to 0. Indeed, two constants B₃, B₄ > 0 can be sorted with the estimates
()
for all integers N ≥ 0, provided that .

At this point, we stress the fact that the generalized expansion of Gevrey type (7) obtained for the solutions y_p(t, z, ɛ) of Equation (2) in the monomial scale are obtained by means of the Ramis–Sibuya theorem (see Theorem (R.S.) in Section 4.2) through precise estimates of some exponential decay for the differences of the consecutive maps y_p+1 − y_p relatively to ɛ on the intersections . These estimates were achieved according to the fact that the Borel–Fourier maps τ₁ ↦ w_p(τ₁, τ₂, m, ɛ) are analytic at τ₁ = 0 in Equation (3). In contrast, for the problem (1) under study, as observed earlier in this introduction, any of the partial Borel–Fourier map appearing in Equation (8) for any admissible direction is not analytic near τ₁ = 0, only on sectors centered at 0. Therefore, no bounds for differences of solutions u(t, z) to Equation (1) for different directions can be rooted out and the recipe using the Ramis–Sibuya theorem fails to be applied. Instead, we introduce a new approach based on a specific splitting of the triple integral (8) defining u(t, z) and on the observation that the partial map can be analytically continued near τ₁ = 0 provided that τ₂ remains on the small disc D_ρ, see Proposition 10. Besides, whereas explicit differential recursions could be provided for the coefficients , n ≥ 0 of the formal expansions (6), no such relations are achieved for the coefficients b_n(t, z), n ≥ 0 of Equation (14). However, explicit equations (displayed in Equation (247)) for b_n, n ≥ 0, can be presented as double truncated q−Laplace, Laplace transforms and inverse Fourier integral of derivatives of the partial Borel–Fourier map at the origin.

2. Setup of the Main Initial Value Problem and an Associated Set of q−Difference-Differential Problems with Homography Action

2.1. Accounts on q−Laplace Transforms of Order k and Fourier Inverse Maps

This concise subsection presents the basic material about q−Laplace transforms and Fourier inverse maps that will be handled to built up the solution of our main problem under study.

Let k ≥ 1 be an integer and set q > 1 a positive real number. We present the definition of a q−Laplace transform of order k as described in our former work [10]. In the construction of this q−analog of the classical Laplace transform of order k, the Jacobi Theta function of order k is defined as the Laurent series

()

for any

plays a prominent role.

We remind that the set of zeros of this analytic function is given by

and is contained on the real line

. The next lower bounds for the Jacobi Theta function attesting its so-called q−exponential growth of order k on a domain bypassing this set of zeros are essential. Let Δ > 0. A constant C_q,k > 0 relying on q, k and independent of Δ can be chosen such that

()

provided that

with |1 + xq^m/k| > Δ for all

Definition 1. Let D_ρ be a disc of some radius ρ > 0 centered at 0 and S_d be an open unbounded sector edged at 0 with bisecting direction in . Let us consider a holomorphic function assumed to be continuous up to the closure and subjected to the bounds

()

for all x ∈ D_ρ ∪ S_d, for some given positive constants K, α > 0, δ > 1, and some integer k ≥ 1. We select some direction

such that

. The q−Laplace transform of order k of f in direction γ is assigned as

()

where

stands for a halfline in direction γ.

Let Δ > 0 be some fixed real number. The integral transform represents a bounded holomorphic function on the domain , for any radius r₁ constrained by

()

and where

()

In the special case

is an entire function with Taylor expansion

conforming to the bounds of Equation (19), its q−Laplace transform of order k, Equation (20) does not depend on the direction

and defines a bounded holomorphic function on

under the restriction of Equation (21), which possesses a Taylor expansion given by the convergent series

The next Banach space of continuous function on with exponential decay was introduced in [11].

Definition 2. Let β, μ be real numbers. We denote E_{(β, μ)} the vector space of continuous functions such that

()

is finite. The space E_{(β, μ)} endowed with the norm ‖.‖_{(β, μ)} becomes a Banach space.

We recall the definition of the inverse Fourier transform acting on the space E_{(β, μ)}.

Definition 3. Let f ∈ E_{(β, μ)} with β > 0, μ > 1. The inverse Fourier transform of f is given by

()

for all

. The function

extends to an analytic bounded function on the strips

()

for all given 0 < β^′ < β.

The next lemma described how the inverse Fourier integral is transformed under the action differential operators and products.

Lemma 1.

(a)
Let f be an element of E_{(β, μ)} for β > 0, μ > 1. Define the function which belongs to the space E_{(β, μ − 1)}. Then, the next identity
()
occurs for all , for any 0 < β^′ < β.
(b)
Take g ∈ E_{(β, μ)} and set the convolution product of f and g
()
Then, ψ belongs to E_{(β, μ)} and moreover, the next equality
()
holds for all , provided that 0 < β^′ < β.

2.2. Layout of the Main Problem

Throughout this subsection, we unveil the principal initial value problem under investigation in this work. It is shaped as follows:

()

for vanishing data u(0, z) ≡ 0, where σ_q;t stands for the q−difference operator acting on t by means of σ_q;tu(t, z) = u(qt, z) for some given real number q > 1.

The set I represents a finite subset of

and d_D, k₁ ≥ 1 are positive integers that are subjected to the next list of technical constraints:

(1)
The inequality
()
holds for all .
(2)
The restrictions
()
are required for all .

The maps Q(X), R_D(X), and

for

are polynomial required to fulfill the next features:

(1)
The degrees of Q and of are constrained by the relation
()
for all .
(2)
We assume the existence of an open sectorial domain with inner radius (respectively outer radius ) given by
()
for some opening , which satisfies the next inclusion
()
for all . Furthermore, the inner and outer radii, together with the aperture of , will be suitably constrained later on in the work.

The coefficients

, are built up through the next procedure. For

, we consider maps

that belong to the Banach space E_{(β, μ)}, for given real numbers β > 0 and μ > 1 constrained to

()

for all

. We introduce the constants

()

for all

on which restrictions will be set in due course of the paper. We define the coefficient

as the inverse Fourier transform

()

for all

, provided that z ∈ H_β. According to Definition 3, the maps

stand for bounded holomorphic functions on the strips

for any prescribed 0 < β^′ < β.

The forcing term is described in terms of the next construction. Let J₁, J₂ be finite subsets of the positive natural numbers

. For all j₁ ∈ J₁, j₂ ∈ J₂, we deem some maps

which appertain to E_{(β, μ)} for β > 0 and μ > 1 given above. We introduce the next polynomial

()

in the variables τ₁,τ₂ with coefficients in E_{(β, μ)}. We bring in the map

()

where

stands for a halfline in some given direction

and

is the negative real axis.

Owing to Definitions 1 and 3, this map F_π(u₁, u₂, z) is well defined provided that

(1)
the variable u₁ belongs to , for any fixed Δ₁ > 0 and radius r₁ > 0 subjected to Equation (21) where k = k₁, for any given α > 0,
(2)
the variable u₂ is not vanishing and obeys the constraint cos(arg(u₂)) < −Δ₂, for some Δ₂ > 0,
(3)
the variable z is kept in the strip for any 0 < β^′ < β.

However, we can further simplify the expression of F_π. Taking heed of Definition 1, we notice that F_π(u₁, u₂, z) turns out to be a polynomial in u₁,

()

whose coefficients

are expressed through sums over J₂ of Laplace transforms in direction π,

()

with bounded holomorphic coefficients on

. Besides, according to the definition of the Gamma function and Cauchy’s theorem, we acknowledge that

()

provided that

with cos(arg(u₂)) < 0. On that account, it follows that F_π(u₁, u₂, z) can be expanded as a polynomial in both variables u₁ and u₂ with bounded coefficients on

, for 0 < β^′ < β. Namely, we get

()

where we define

()

for all j₁ ∈ J₁ and j₂ ∈ J₂. At last, we configure the forcing term f(t, z) as the logarithmic type function

()

Here, log(t) stands for the principal value of logarithm, namely

provided that −π < arg(t) < π. Furthermore, we observe that

()

for some Δ₂ > 0, whenever t ∉ (−∞, 0] and close enough to 0.

In the particular case J₁ = {j₁} and J₂ = {j₂} for some positive integers j₁, j₂ ≥ 1, we make the noteworthy remark that the solution u(t, z) of the linear main Equation (29) actually solves a special nonlinear q− difference-differential equation with polynomial coefficients in some positive rational power of time t stated in Equation (50). Indeed, let the forcing term f(t, z) have the particular shape

()

where

is given by the expression (44). By direct computation, we check that the forcing term

satisfies the next nonlinear ordinary differential equation with polynomial coefficients in

()

Let us recast the main Equation (29) in the form

()

where the q−difference-differential operator P is polynomial in t, with bounded holomorphic coefficients in z on the strip

, for 0 < β^′ < β. The combination of Equations (48) and (49) gives rise to the next nonlinear equation

()

2.3. A Set of Related q−Difference-Differential Equations with an Homography Action

In this subsection, the main problem is embedded in a set of auxiliary problems which comprise three independent complex variables, which will be the object of study in the forthcoming sections.

We seek for solutions u(t, z) to Equation (29) for prescribed vanishing initial data at t = 0 of the form

()

for some expression U_π(u₁, u₂, z) in the three independent variables u₁, u₂, and z.

The next computations hold for any given rational number h > 0,

()

where

(1)
the dilation acts on U_π relatively to u₁ through ,
(2)
the homography is applied on U_π with respect to the variable u₂ by means of

()

As a result, it follows that the expression u(t, z) (formally) solves Equation (29) under the condition u(0, z) ≡ 0 if the expression U_π(u₁, u₂, z) fulfills the next equation

()

under the constraint U_π(0, 0, z) ≡ 0. Later on, we plan to build a genuine solution to Equation (29), and in order to investigate its asymptotic expansion in some particular scale described in Section 4.3, we are required to complement the above single Equation (54) by a family of auxiliary problems stated underneath.

For any given direction

with d₂ ≠ π (modulo 2π) and given positive real radius a > 0, we define a new forcing term

()

where

stands for a segment of length a > 0 in direction d₂ and

is the halfline appearing in Equation (39). Owing to Definition 1, we notice that the map

does not rely on the direction d₁. However, it hinges on the direction d₂ and radius a > 0. We display the next problem

()

for given vanishing initial data

3. Analytic Solutions to the Associated Set of q−Difference and Differential Problems under Homography Action

In this section, we intend to exhibit analytic solutions to the problems Equations (54) and (56) we came up with in Section 2.3.

3.1. Profile of the Analytic Solutions and Joint Convolution q−Difference Equations

We search for a solution to Equation (54) (respectively Equation (56) for d₂ ≠ π modulo 2π) in the form of a double q−Laplace, Laplace transform and inverse Fourier integral with the shape

()

respectively

()

Here, it is assumed (this fact will be justified later on in the work) that the so-called Borel–Fourier map

appertains to a Banach space labeled

, which consists in functions with so-called q−exponential growth of order k₁ w.r.t τ₁, exponential growth in τ₂ and exponential decay relatively to the mode m. This space is described in the next,

Definition 4. We consider the constants β, μ, k₁, a as prescribed in Section 2. Let α, ν > 0 and ρ > a, δ > 1 be real numbers. We set as an unbounded sector edged at 0 with bisecting direction . We introduce the open half-strip

()

for some given real width η₂ > 0. We denote

the vector space of all

valued continuous maps (τ₁, τ₂, m) ↦ h(τ₁, τ₂, m) on the domain

, holomorphic w.r.t (τ₁, τ₂) on the product

, such that the norm

()

is finite. The vector space

equipped with the norm

represents a Banach space.

Our main objective is to establish some convolution q−difference equation that the Borel–Fourier map is asked to obey. On the way, we need some additional features on the q−Laplace transforms under multiplication by a monomial and action of q−difference operators. These properties have already been discussed in our past work [10]. Besides, we describe the action of the homography relatively to the variable u₂ on both expressions Equations (57) and (58).

Lemma 2. Let the map supposed to belong to the Banach space . Then, the next identities hold.

(1)
For prescribed integers l₀, l₁ ≥ 0, the q−difference operator acts on the integral representations Equations (57) and (58) through the equations

()

and

()

(2)
For a given rational number h > 0, the homography applies on the triple integrals Equations (57) and (58) by means of

()

along with

()

In line with the above technical lemma together with Lemma 1, the next statement follows:

Lemma 3. The map solves Equation (54) under the constraint and obeys Equation (56) for d₂ ≠ π (modulo 2π) with vanishing data if the Borel–Fourier map fulfills the next convolution q−difference equation

()

provided that

, τ₂ ∈ H_π ∪ D_ρ, and

3.2. Solving the Convolution q−Difference Equation (65) on Unbounded Sectors and Half Strips

In the course of this subsection, we prove the existence and unicity of a solution to the convolution q−difference reached in Lemma 3.

Our scheme consists of reorganizing Equation (65) as a fixed point equation (displayed later on in Equation (119)). On the way, we are asked to divide our equation by the next Fourier mode depending map with two complex variables

()

provided that

and τ₂ ∈ H_π ∪ D_ρ. An essential factorization of the above map is provided in the next lemma.

Lemma 4. For a convenient choice of the inner radius , outer radius and aperture of set up in Equation (34), one can distinguish an unbounded sector edged at 0 with suitable bisecting direction along with an appropriate strip H_π and a small radius ρ for which the next splitting of the map P_m(τ₁, τ₂) holds. Let written in the factorized form

()

for some radius r > 0 and complex number

with

. Let us take τ₂ ∈ H_π ∪ D_ρ. Then, one can decompose τ₂ in the form

()

for some well chosen complex number

(depending on

and m and which remains bounded relatively to m), for some ψ ≠ 0, close to 0 and some s ≥ −A (for some fixed constant A > 0). With the above factorizations Equations (67) and (68), one can express the map P_m in the form of a non-vanishing product

()

for given

and τ₂ ∈ H_π ∪ D_ρ.

Proof. We choose appropriately the sectorial domain given in Section 2.2 and select an unbounded sector edged at 0 with bisecting direction d₁ chosen in a way that the next constraint

()

holds for all

, all

for some small positive numbers 0 < α₁ < α₂. Let

be given. We can factorize it in the form of Equation (67). We set

()

Notice that

remains bounded and penned in a small domain we denote

which is located at some small positive distance of the real axis when m spans the real numbers according to the condition Equation (34) imposed.

In the next step, we select the strip H_π and the disc D_ρ in a way that

()

As a result, when one takes some element τ₂ ∈ H_π ∪ D_ρ, we can write it in the form Equation (68) for some real number s > −A for some A > 0 and some real number ψ ≠ 0 that can be chosen close to 0.

By construction of , we get in particular that

()

In consequence of the combined factorizations Equations (67) and (68) together with the above identity Equation (73), the next computations hold

()

which is exactly the announced expression Equation (69). In particular, this product is nonvanishing since

, for all

owing to Equation (34) and considering that ψ ≠ 0 but close to the origin, the piece enclosed by brackets in Equation (69) cannot vanish.

Let us consider the next linear map

()

In the next proposition, it is stated that the map

stands for a shrinking map on some fittingly chosen ball of the Banach space discussed in Definition 4.

Proposition 1. We select the sectorial domain , the unbounded sector together with the strip H_π and the disc D_ρ as in Lemma 4. Then, provided that the constants displayed in Equation (36) are small enough, for , an adequate radius ϖ > 0 can be chosen for which the map enjoys the next two properties

(1)
The inclusion
()
is granted, where denotes the closed ball of radius ϖ centered at 0 in the space .
(2)
The 1/2−Lipschitz condition
()
holds for all .

Proof. We first aim our attention to the inclusion Equation (76). Let us prescribe some real number ϖ > 0 and take some element ω(τ₁, τ₂, m) of subjected to the condition

()

We plan to disclose norm estimates for each piece of the map

. We first focus on the norm upper bounds for the elements involved in the sum over I. The next technical lemma is crucial in this respect.

Lemma 5. Under the imposed constraints Equations (30) and (31) together with Equations (32), (34), and (35), one can find a constant C₁ > 0 such that

()

for all

Proof. According to Definition 4, we next upper bounds hold for the element ω,

()

for all

, all τ₂ ∈ H_π ∪ D_ρ,

. We deduce first upper bounds

()

whenever

, τ₂ ∈ H_π ∪ D_ρ, and

. The resulting bounds of Equation (79) will be reached after several steps of computations. Namely,

(1)
We provide upper bounds for the function

()

for

. Since

are polynomials, we get a constant

with

()

for all

. Besides, owing to the assumption (34), a constant Q > 0 can be pinpointed with the lower bounds

()

for all

and from the definition of the constants

, we know that

()

for all

. The collection of bounds of Equations (83), (84), and (85) together with the triangular inequality |m| ≤ |m − m₁| + |m₁| enable the next estimates

()

At last, according to Lemma 2.2 from [11], we call to mind that the quantity

()

is finite under the assumption of Equations (32) and (35). Therefore, a constant C_1.1 > 0 can be singled out with

()

for all

(2)
We focus on upper estimates for the quantity

()

provided that

with |τ₁| > r₁ for some fixed real number r₁ > 1. We need to perform the next expansions

()

together with

()

Owing to the freshman classical limit

and equivalence relation log(1 + x) ~ x as x tends to 0, we reach a two constants A₁, A₂ > 0 with

()

provided that |τ₁| > r₁ > 1. Furthermore, since x ↦ log²⁡(x) and x ↦ log(x) are both increasing maps on [1, +∞), we observe the inequalities

()

whenever |τ₁| > r₁ > 1. From the two expansions Equations (90) and (91) and the bounds of Equations (92) and (93) together with the assumption Equation (30), we arrive at the next estimates

()

provided that

with |τ₁| > r₁, for some constant C_1.2 > 0.

(3)
We supply upper bounds for the quantity

()

for

with |τ₁| > r₁ and

where r > 0,

, ψ ≠ 0 close to 0 and s > −A for some constant A > 0, according to the decompositions Equations (67) and (68). We recast

in the form

()

Taking heed of our assumption Equation (31), we obtain a constant C_1.3 > 0 with

()

for all r > r₁ and s > −A, for ψ ≠ 0 close to 0. Furthermore, a constant C_1.4 > 0 can be singled out with

()

as long as r > r₁ and s > −A, for ψ ≠ 0 close to 0. From Equations (97) and (98), we deduce a constant C_1.5 > 0 such that

()

whenever

with |τ₁| > r₁ and all τ₂ ∈ H_π ∪ D_ρ.

(4)
We establish bounds for the quantity displayed in Equation (89) provided that with |τ₁| ≤ r₁, where r₁ > 1 has been fixed in (2). A mere observation yields a constant C_1.6 > 0 with

()

for all

with |τ₁| ≤ r₁.

(5)
We present bounds for the piece

()

for

with |τ₁| ≤ r₁ and

, where r > 0,

, ψ ≠ 0 close to 0 and s > −A for some constant A > 0, according to the decompositions Equations (67) and (68). We rearrange

as follows:

()

Bearing in mind the condition (31), we get a constant C_1.7 > 0 with

()

provided that 0 < r ≤ r₁ and s > −A, for ψ ≠ 0 close to 0. On the other hand, a constant C_1.8 > 0 can be set with

()

as long as 0 < r ≤ r₁ and s > −A, for ψ ≠ 0 close to 0. Due to Equations (103) and (104), a constant C_1.9 > 0 can be picked out such that

()

for all

with |τ₁| ≤ r₁ and all τ₂ ∈ H_π ∪ D_ρ.

(6)
As a consequence of the list of estimates Equations (94), (99), (100), and (105), we obtain a constant C_1.10 > 0 with

()

for

and

where r > 0,

, ψ ≠ 0 close to 0 and s > −A for some constant A > 0, according to the decompositions Equations (67) and (68).

In conclusion, on the basis of the factorization of Equation (69) for the map P_m(τ₁, τ₂) together with the bounds of Equations (88) and (106) combined with the bounds of Equation (81), we arrive at the next inequality

()

for all

, all τ₂ ∈ H_π ∪ D_ρ. Notice that this last inequality is tantamount to the awaited bounds of Equation (79) for the constant

()

We need control on the norm of the last term of related to the forcing term of Equation (65).

Lemma 6. There exists a constant such that

()

Proof. In view of the factorization (69) and the definition (38) of , we notice that

()

for all

and τ₂ ∈ H_π ∪ D_ρ for which the splittings Equations (67) and (68) hold, and all

. Besides, by definition of

, constants

can be found such that

()

for all

. Furthermore, we can pinpoint a constant

for which

()

hold where

and τ₂ ∈ H_π ∪ D_ρ with the decompositions Equations (67) and (68). As a result, combining Equations (110), (111), and (112) gives rise to the next upper estimates

()

for all

and τ₂ ∈ H_π ∪ D_ρ, where

()

keeping in mind that

, for k = 1, 2. At last, it remains to notice that the due inequality of Equation (109) results from Equation (113) by taking heed of Definition 4.

We select the constant ϖ > 0 suitably together with the constants , for , taken close enough to 0 in a way that the next inequality

()

holds where C₁ > 0 appears in Lemma 5 and

stems from Lemma 6. Eventually, the expected inclusion of Equation (76) prompts from the bounds of Equations (79) and (109) under the restriction Equation (115).

We discuss the second item addressing the shrinking feature Equation (77). We take two elements ω₁, ω₂ in the closed ball from whose radius ϖ > 0 has been prescribed in the first item Equation (76). According to Lemma 5, under the conditions Equations (30), (31), (32), (34), and (35) listed in Section 2.2, the next inequality

()

holds for the constant C₁ > 0 introduced in Lemma 5. We set the constants

, for

, small enough allowing the next inequality

()

to hold. The Lipschitz property Equation (77) is a straight consequence of Equation (116) under the requirement Equation (117).

In conclusion, we properly choose the constants , and the radius ϖ > 0 in order to impose both constraints Equations (115) and (117) at once, which triggers the two properties Equations (76) and (77) for the map .

In the forthcoming proposition, we provide a solution to the convolution q−difference Equation (65) established in Lemma 3.

Proposition 2. Let us prescribe the sectorial domain , the unbounded sector together with the strip H_π and the disc D_ρ as in Lemma 4. Then, the constants defined in Equation (36) and a constant ϖ > 0 can be fittingly chosen in a manner that a unique solution to the convolution q−difference Equation (65) can be built up in the space under the condition

()

Proof. We select ϖ > 0 as in Proposition 1. We mind the closed ball in the Banach space which represents a complete metric space for the distance deduced from the norm. Proposition 1 states that induces a contractive map from the metric space into itself. According to the classical Banach fixed point theorem, it follows that owns a unique fixed point inside the ball , we denote . It means that

()

for all

, τ₂ ∈ H_π ∪ D_ρ, and

. By transferring the term

()

from the right to the left handside of Equation (65) and dividing the resulting equation by the map P_m(τ₁, τ₂) displayed in Equation (66), we observe that Equation (65) can be rearranged into the fixed point Equation (119). On that account, the unique fixed point

obtained in

precisely solves Equation (65), which yields Proposition 2.

3.3. Analytic Solutions to the Auxiliary Equations (54) and (56)

In the next proposition, we craft analytic solutions to the associated set of q−difference and differential problems under the action of homographic maps established in Section 2.3.

Proposition 3. The sectorial domain , the unbounded sector together with the strip H_π and the disc D_ρ are prescribed as in Lemma 4.

(1)
We define the map
()
where the Borel–Fourier map is built up in Proposition 2 and solves the convolution q−difference Equation (65). The map Equation (119) boasts the next two qualities
- (i)
  It defines a bounded holomorphic function on the product for some given Δ₁ > 0, where stands for the set Equation (22) and is a disc centered at 0 with radius subjected to the constraint
  ()
  and 0 < β^′ < β. Besides, U_2,π represents a bounded sector edged at 0 with bisecting direction π with radius R₂ > 0, submitted to the next condition: there exists some real number Δ_2,π > 0 with
  ()
  for all u₂ ∈ U_2,π, where 0 < R₂ < Δ_2,π/ν, for ν > 0 fixed in Definition 4.
- (ii)
  It solves the auxiliary Equation (54) for prescribed initial data .
(2)
For a direction d₂ ≠ π (modulo 2π), we shape the map
()
where is the Borel–Fourier map mentioned in the above item. The map Equation (124) enjoys the next two properties:
- (i)
  It represents a bounded holomorphic function on the product , for the domain , disc and constant 0 < β^′ < β given in the first item. Furthermore, stands for a bounded sector centered at 0 with bisecting direction d₂ and with radius R₂ chosen as in the first item and subjected to the next restriction: some positive real number can be found with
  ()
  for all .
- (ii)
  It obeys the auxiliary Equation (56) for given vanishing initial data .

Proof. We discuss the first item. We parametrize and τ₂ ∈ L_π in the form and for r₁, r₂ ≥ 0. Then, owing to Equations (18) and (118), we get

()

for all

with

for all r ≥ 0 and u₂ ∈ U_2,π. In order to provide upper bounds for the right-hand side of Equation (126), we propose the next alternative.

Assume that 0 ≤ r₁ < 1 with as above under the constraint |u₁| ≤ R₁. Then, one can single out a constant such that

()

for all 0 ≤ r₁ < 1 with

Assume that r₁ ≥ 1 and for a radius R₁ > 0 under the constraint (122). The next three expansions are useful. Namely,

()

together with

()

and

()

Since log(1 + x) ~ x holds as x is close to 0 and owing to the classical limit

, we get from Equations (129) and (130) two constants M_δ,1, M_δ,2 > 0 with

()

for all r₁ ≥ 1. As a result, we get from the computation Equation (128) and bounds of Equation (131) that

()

At last, from the assumption (122) and requirement |u₁| ≤ R₁, the next bounds

()

are deduced from Equation (132).

On the other hand, taking heed of Equation (123), we observe that

()

provided that

, where 0 < β^′ < β and

according to the claim that |u₂| < R₂ < Δ_2,π/ν.

As a consequence of the above bounds of Equation (127), along with Equations (133) and (134), we deduce that the map is well defined and represents a bounded holomorphic function on the product under the above requirements Equations (122) and (123).

Recall that the Borel-Fourier map has been constructed as a solution of the associated convolution q−difference Equation (65) in Proposition 2. From Lemma 3, we deduce that obeys the auxiliary Equation (54) on the domain for prescribed initial data .

We turn to the second item. Let and be parametrized as follows and with r₁ ≥ 0, 0 ≤ r₂ ≤ a. Bearing in mind Equations (18) and (118), we obtain a constant ϖ > 0 such that the next inequality

()

holds provided that

and

. According to Equation (125), we notice that

()

under the restriction |u₂| < R₂. By dint of the upper bounds of Equation (127) in a row with Equations (133), (134), and (136), we acknowledge the fact that

is bounded and stands for a holomorphic map on the product

under the assumptions (122) and (125). Since the Borel–Fourier map

solves the convolution q−difference Equation (65) as shown in Proposition 2, we deduce from Lemma 3 that

conforms the auxiliary Equation (56) on the domain

for given vanishing initial data

4. Construction of a Holomorphic Solution to the Main Initial Value Problem (21) and Its Gevrey Asymptotic Expansion Relatively to Complex Time t in Logarithmic Scale

4.1. A Finite Set of Genuine Solutions to Related Initial Value Problems

We restate the definition of a good covering in as described in the textbook [12], Section XI-2.

Definition 5. Let ς ≥ 2 be an integer. A set of bounded sectors edged at 0 is deemed with the next three attributes

(1)
Any two consecutive sectors U_p and U_p+1 have non empty intersection U_p∩U_p+1, for 0 ≤ p ≤ ς − 1, where the convention U_ς = U₀ is assumed.
(2)
The intersection of any three sectors U_p∩U_q∩U_r is reduced to the empty set for all distinct nonnegative integers p, q, r less than ς − 1.
(3)
The union covers some punctured neighborhood of 0 in .

Such a set is tagged a good covering in .

The notion of fitting a set of sectors is discussed in the next definition.

Definition 6. Let ς ≥ 2 be an integer. A finite set of bounded sectors is minded with the next three constraints.

(1)
For each 0 ≤ p ≤ ς − 1, the sector is edged at 0, with bisecting direction and is subjected to the condition that some real number can be singled out with
()
for all .
(2)
There exists an index p₁ ∈ {1, …, ς − 1} with . All the sectors , 0 ≤ p ≤ ς − 1 have the same radius R₂ which obeys the restriction
()
where Δ_2,π > 0 is introduced in the above item and ν > 0 is declared in Definition 4.
(3)
The set forms a good covering in in the sense of Definition 5.

A set endowed with the above three features is called a fitting set of sectors.

In the oncoming proposition, we exhibit analytic solutions to the auxiliary problems Equations (54) and (56), where the directions d₂ span the set of bisecting directions of some fitting set of sectors. Furthermore, sharp estimates of their consecutive differences are provided, which are essential in the study of their asymptotic expansions in the variable u₂ that will be described in Section 4.2.

Proposition 4. Let the sectorial domain , the unbounded sector together with the strip H_π and the disc D_ρ be arranged as in Lemma 4. Consider a fitting set of sectors and assign a radius a with 0 < a < ρ. Then, provided that the constants are taken close enough to 0 in accordance with the requirements of Proposition 2, the properties described in the forthcoming three items hold.

(1)
For each p ∈ {0, …, ς − 1}∖{p₁} (where p₁ stems from Definition 6.2) the equation
()
where the forcing term is given by the triple integral Equation (55), possesses a bounded holomorphic solution on the domain , where stands for the set (22), for a radius R₁ > 0 fulfilling (122), which observes the condition . Furthermore, the map is embodied in a Fourier inverse and a double q−Laplace, Laplace transform
()
where the Borel–Fourier map belongs to the Banach space (introduced in Definition 4) constrained to the bounds of Equation (118).
(2)
The equation
()
with forcing term F_π is displayed in Equation (39) and expressed as a polynomial in Equation (43), holds a bounded holomorphic solution on the domain , where the set and radius R₁ are given in the above item, under the vanishing condition . In addition, the map is expressed through a Fourier inverse and a double q−Laplace, Laplace transform
()
where the Borel–Fourier map is described in the former item.
(3)
The neighboring differences of the maps are controlled by the next bounds. For all 0 ≤ p ≤ ς − 1, two constants M_p,1, K_p,1 > 0 can be found such that
()
for all , all , provided that for a well chosen radius . Here we adopt the convention that .

Proof. The first two items are direct corollaries of the statement of Proposition 3 and the definition of a fitting set of sectors chosen at the onset of Proposition 4.

We focus on the third item, which demands more labor and hinges on path deformations arguments. We distinguish two different situations.

Case 1. Let p = p₁ or p = p₁ − 1. We discuss only the subcase p = p₁ since the other alternative p = p₁ − 1 can be treated in a similar manner. By construction, we notice that (modulo 2π). According to Proposition 2, for any prescribed and , the partial map is analytic on the union H_π ∪ D_ρ. As a result, the oriented path can be bent into the union of

(1)
the halfline ,
(2)
the arc of circle ,

and the classical Cauchy’s theorem enables the difference

to be reorganized as a sum of two contributions. Namely,

()

for all

, all

and

. We need to control the first piece of Equation (144)

()

Drew on the bounds of Equations (126) and (127) together with Equations (133) and (134), we split the halfline

in the union of two segments

and

and we are reduced to provide bounds for the next two quantities J_1.1 and J_1.2 for

()

where

()

and

()

Indeed,

()

and

()

Now, we set

for some real number

. Hence,

()

provided that

. As a result of Equations (149), (150), and (151), we deduce from the splitting Equation (146) that

()

for all

, all

and

, where

()

In the next step, we display bounds for the second piece of Equation (144)

()

According to Definition 6.1 of a fitting set of sectors, we notice that the lower bounds

()

for all

whenever the angle θ belongs to

. By breaking up the halfine

into the segments

and

, similar computations as above yield the bounds

()

for all

, all

, as long as

In conclusion, the decomposition of Equation (144), along with the two upper bounds of Equations (152) and (156), beget the estimates of Equation (143) under the assumption that p = p₁.

Case 2. Assume that p ∉ {p₁ − 1, p₁}. We observe that both directions and are not equal to π modulo 2π. Owing to Proposition 2, for any fixed and , the partial map is analytic on the disc D_ρ. On these grounds, we can deform the oriented path into a single arc of circle

()

and rewrite the difference

as a single triple path integral

()

for all

, all

and

. Upper bounds are asked for the quantity

()

Definition 6 of fitting sets of sectors allows the next lower bounds

()

to hold for all

, whenever the angle θ is taken in

. Using the partition of the halfline

in two segments

and

, comparable estimates as the ones performed in Case 1 give rise to the next bounds

()

for all

, all

, provided that

, where

is given by the expression (153).

In brief, the recast expression (158) coupled with the bounds of Equation (161) prompts the awaited estimates of Equation (143) under the assumption that p ∉ {p₁ − 1, p₁}.

4.2. Gevrey Asymptotic Expansions for the Bounded Holomorphic Solutions to the Family of Auxiliary Problems (139) and (141)

In the next proposition, asymptotic expansions of Gevrey type are achieved for the maps , that are displayed in Proposition 4, relatively to the variable u₂.

Proposition 5. For the constants d₁, Δ₁, R₁ and β^′ fixed in Proposition 4, we denote the Banach space of valued bounded holomorphic functions on the product endowed with the sup norm. Then, for all 0 ≤ p ≤ ς − 1, the partial maps , viewed as bounded holomorphic maps from the bounded sector into , share a common formal power series

()

with coefficients G_n, n ≥ 0, that belong to

, as Gevrey asymptotic expansion of order 1 on

. It means that, for each 0 ≤ p ≤ ς − 1, two constants K_p,2, M_p,2 > 0 can be chosen in a way that the next error bounds

()

hold for all integers N ≥ 0, all

, whenever

and

Proof. In the proof, we apply the next result, known as the Ramis–Sibuya theorem, and we rephrase for the sake of completeness and clarity for the reader (see Lemma XI-2-6 in [12]).

Theorem (R.S.). Let be a Banach space over the field of complex numbers and let be a good covering in as outlined in Definition 5. For all 0 ≤ p ≤ ς − 1, we consider holomorphic functions that enjoy the next two features

(1)
The maps G_p are bounded on U_p for all 0 ≤ p ≤ ς − 1.
(2)
The difference Θ_p(u) = G_p+1(u) − G_p(u) stands for a holomorphic map on the intersection Z_p = U_p+1∩U_p which is exponentially flat of order k, for some integer k ≥ 1, meaning that one can select two constants C_p, A_p > 0 for which
()
holds provided that u ∈ Z_p, for all 0 ≤ p ≤ ς − 1. By convention, we set G_ς = G₀ and U_ς = U₀.

Then, a formal power series

with coefficients G_n belonging to

can be singled out, which is the common Gevrey asymptotic expansion of order 1/k relatively to u on U_p for all the maps G_p, for 0 ≤ p ≤ ς − 1. It attests that two constants K_p, M_p > 0 can be chosen with the result that the error bounds

()

hold for all integers N ≥ 0, all u ∈ U_p, all 0 ≤ p ≤ ς − 1.

For each 0 ≤ p ≤ ς − 1, we introduce the map

set as

()

In view of Proposition 4, we acknowledge that

(1)
The set of sectors forms a good covering in owing to Definition 6.3.
(2)
For each 0 ≤ p ≤ ς − 1, the map G_p is bounded holomorphic on the sector .
(3)
For each 0 ≤ p ≤ ς − 1, the difference Θ_p(u₂) = G_p+1(u₂) − G_p(u₂) suffers the bounds
()
for the constants M_p,1 and K_p,1 displayed in Equation (143), provided that .

Thereupon, Claims 1 and 2 of Theorem (R.S) are matched for the family of maps with the constant k = 1. The existence of the formal power series of Equation (162), which represents the collective Gevrey asymptotic expansion of order 1 relatively to u₂ on for all the maps G_p, 0 ≤ p ≤ ς − 1 follows. As a result, the error bounds of Equation (163) are warranted.

4.3. Statement of the First Main Result

In this subsection, a bounded holomorphic solution to our main initial value problem (29) is shaped. This solution is favored with an asymptotic expansion in some logarithmic scale that reveals to be of Gevrey type. The next theorem represents the first main achievement of our work.

Theorem 1. Let the sectorial domain , the unbounded sector together with the strip H_π and the disc D_ρ be duly prescribed as in Lemma 4. Then, assuming that the constants are in the vicinity of 0 as specified by the requirements of Proposition 2 and that the radius R₁ > 0 is close enough to 0, the equation

()

has a bounded holomorphic solution (t, z) ↦ u(t, z) on the domain

for vanishing initial data u(0, z) ≡ 0. In addition, the map u(t, z) can be expressed as a triple integral comprising a Fourier inverse, a q−Laplace and Laplace transforms

()

where the Borel–Fourier map

originates from the Banach space

(see Definition 4) and is restrained to the bounds of Equation (118).

The function u(t, z) enjoys a generalized asymptotic expansion of Gevrey type in a logarithmic scale as t tends to 0. More precisely, one can single out a formal series

()

with bounded holomorphic coefficients G_n(t, z) on the domain

, which stands for an asymptotic expansion of Gevrey order 1 in the scale of logarithmic functions

of the map u(t, z) with respect to t on the domain

. In other words, two constants

can be found with the aim that the next error bounds

()

hold for all integers N ≥ 0, all

, provided that

Proof. We select a fitting set of sectors and we take the index p = p₁ for which according to Definition 6.2. By definition of the principal value of the logarithm , for arg(t) ∈ (−π, π), whenever , we check that

()

as long as

, provided that we take R₁ > 0 sufficiently close to 0, where

has been disclosed in the third item of Proposition 4. We define

()

where the map

is described in the second item of Proposition 4. By construction of

, we ascertain that u(t, z) represents a bounded holomorphic function on the product

Besides, according to the second item of Proposition 4, we know that the map stands for a solution to Equation (141) on the domain . On the basis of the computations made in Section 2.3, we deduce that the map u(t, z) solves the main Equation (29) on the domain , constrained to the initial value condition u(0, z) ≡ 0.

At last, the asymptotic expansion property of Equation (171) of the map u(t, z) is a direct offspring of the expansion of Equation (163) for the particular case p = p₁, where u₁ is set to be the time variable t and the variable u₂ is merely replaced by the logarithmic function 1/log(t) for .

4.4. Computational Features Related to the Formal Power Series (162)

In this subsection, we establish that the formal series (162), which represents the asymptotic expansion of Gevrey type for the holomorphic maps actually solve some functional partial differential equation. On the journey, we notice that its coefficients G_n, n ≥ 0 fulfill some handy recursion relations that might be of interest for concrete applications.

Proposition 6. The formal power series

()

with coefficients G_n, n ≥ 0 in the space

, conforms the next functional partial differential equation

()

In addition, the coefficients G_n, n ≥ 0 satisfy the recursion relations (189) and (190).

Proof. We depart from Equation (141) recast in the form

()

provided that

, u₂ ∈ U_2,π, and

. We remind the reader of the next useful classical result, which relates the coefficients of an asymptotic expansion of a holomorphic map f to its high-order derivatives.

Proposition. (See [4], Proposition 8, p. 66) Let be a holomorphic map from a bounded open sector G centered at 0 into a complex Banach space endowed with a norm . The following two statements are equivalent

(1)
There exists a formal power series with coefficients f_n in subjected to the next feature. For all closed subsector S of G centered at 0, there exists a sequence (c(N, S))_N≥0 of positive real numbers such that
()
for all z ∈ S, all integers N ≥ 1.
(2)
All derivatives of order n, f⁽ⁿ⁾(z) are continuous at the origin and there exists a sequence of elements in such that
()
for all integers n ≥ 0.

As a result of the above proposition, we deduce from the asymptotic expansion (163) in the particular case p = p₁ (meaning that

) the next limits

()

for all integers m ≥ 0. On the basis of the above limits, in order to reach recursion relations for the coefficients G_m, m ≥ 0, our strategy consists in searching for recursion relations for the related mth derivatives of the map

relatively to u₂. On the way, we need to take the mth derivative with respect to u₂ of the left and right handside of Equation (176). However, Equation (176) involves the composition of

with explicit homographic maps, and we are asked to explicitly compute their higher order derivatives. In order to overcome this difficulty, we will apply a rule to evaluate high-order derivatives of compositions of functions, which has been introduced in [13] and is suitable for Gevrey estimates. This rather new identity allows us to avoid computations with the cumbersome combinatorial classical Faa-Di-Bruno formula and enables us to present very practical recursion relations. Indeed, we recall this higher order chain rule (Theorem 2.1 in [13]) under stronger assumptions in the framework of holomorphic functions on open sets in

Lemma 7. Let D, G be open sets in . Let g : D → G and be holomorphic functions. Then, the nth order derivative of the composite function is given by the equation as follows:

()

for all integers n ≥ 1 and x ∈ D.

In the next lemma, we perform an auxiliary computation, which entails the homographic maps appearing in the main Equation (176).

Lemma 8. For any integer l ≥ 0, we set

()

Then, for all integers n, j ≥ 1 with n − j ≥ 0, the next identity

()

holds for all u₂ ∈ U_2,π, with the convention that j(j + 1) ⋯ (n − 1) = 1 when j = n.

Proof. Direct computations show that

()

and hence

()

for all u₂ ∈ U_2,π. We deduce that

()

for all u₂ ∈ U_2,π and all integers j ≥ 1. It follows from Equation (185) that

()

which coincides with Equation (182) in the case j = n ≥ 1 under the convention that j(j + 1) ⋯ (n − 1) = 1. On the other hand, when n − j ≥ 1, we deduce from Equation (185) that

()

which yields the awaited identity Equation (182) by setting h = 0 in Equation (187).

On the ground of the above lemmas and based on Equation (176), we can derive some recursion relation on the sequence of mth derivatives

with respect to u₂. Namely,

()

for all m ≥ 1, all

, all u₂ ∈ U_2,π and all

In the next step, we let u₂ tend to 0 on the sector U_2,π in both identities Equations (176) and (188). According to the limits (179) and bearing in mind that the maps

and G_m(u₁, z) are holomorphic relatively to

, we get the next recursion relations for the coefficients G_m, m ≥ 0. Namely,

()

together with

()

for all m ≥ 1, all

and

In the last part of the proof, we show that the formal power series (174) obeys the functional Equation (175). Our approach hinges on the next technical lemma, where the Taylor expansion of the composition of the formal series (174) with some homographic map is explicitly computed.

Lemma 9. Let l ≥ 0 be an integer. The next formal Taylor expansion

()

holds, for all

and

Proof. By mere composition, we notice that

()

On the other hand, the geometric series allows to write

()

and taking its derivative of order n ≥ 0 with respect to u₂ yields the expansion

()

with the notation h(h − 1) ⋯ (h − (n − 1)) = 1 if n = 0 and h(h − 1) ⋯ (h − (n − 1)) = h if n = 1, for all h ≥ n. From Equation (194), for all integers n ≥ 1, we deduce the next identity

()

As a result of Equations (192) and (195), we deduce that

()

Besides, by straight calculus, we observe that

()

for all m ≥ 1 and 1 ≤ j ≤ m. Eventually, the combination of Equations (196) and (197) yields the awaited formal Taylor expansion (191).

According to the fact observed in Equation (43) that the map F_π defines a polynomial in the variable u₂, it follows that its Taylor expansion

()

is convergent (and actually a finite sum) near the origin with respect to u₂, for all

and

At the very end of the proof, we observe by plugging the expansions Equations (191) and (198) into Equation (175) that the series formally solves Equation (175) if its Taylor coefficients G_m, m ≥ 0 fulfill the recursion relations Equations (189) and (190) which has been shown to hold. Proposition 6 follows.

5. Fine Structure of Gevrey/q−Gevrey Asymptotic Expansions in Combined Power and Logarithmic Scales for the Holomorphic Solution to the Initial Value Problem (29)

5.1. Solving the Convolution q−Difference Equation (65) on Some Neighborhood of the Origin

In order to study Equation (65) in the Borel space near the origin in and Fourier space on , we introduce the next Banach space.

Definition 7. Let β, μ, ρ > 0 be real numbers. For a given real number b > 0, we denote E_{(b, ρ, β, μ)} the vector space of all continuous valued functions (τ₁, τ₂, m) ↦ h(τ₁, τ₂, m) on , holomorphic with respect to (τ₁, τ₂) on D_b × D_ρ, such that the norm

()

is finite. The vector space E_{(b, ρ, β, μ)} endowed with the norm ‖.‖_{(b, ρ, β, μ)} is a Banach space.

We plan to solve the next convolution q−difference equation

()

provided that τ₁ ∈ D_b, τ₂ ∈ D_ρ, and

, with some function ω in the Banach space (E_{(b, ρ, β, μ)}, ‖.‖_{(b, ρ, β, μ)}).

In preparation for achieving our goal, we rearranged Equation (200) as a fixed point equation (disclosed later on in Equation (225)). Along the road, we need to divide our equation by the map P_m(τ₁, τ₂) displayed in Equation (66) whenever τ₁ ∈ D_b, τ₂ ∈ D_ρ, and the mode m belongs to . Lower bounds for the map P_m are provided in the next lemma.

Lemma 10. Let the inner radius , outer radius and aperture of introduced in Section 2.2 be chosen as in Lemma 4. Let ρ > 0 be the radius fixed in Lemma 4. Then, for a proper choice of radius b > 0, taken close enough to 0, one can find a constant with

()

for all τ₁ ∈ D_b, all τ₂ ∈ D_ρ, all

Proof. Take a fixed . We introduce the complex number

()

Observe that

remains bounded and parked in a small domain we denote

which is located at some small positive distance of the origin, when m varies within the real numbers, owing to the requirement (34). We select the radius b > 0 accordingly to the condition

()

Now, let us take an arbitrary complex number τ₂ ∈ D_ρ. We decompose it in the form

()

for some real numbers

close to 0 for

given above. By construction of

in Equation (202), the next identity

()

holds. Select some arbitrary τ₁ ∈ D_b. We split it in a factorized form

()

for some angle

and radius

with the constraint

. The combined splitting of Equations (204) and (206), together with the identity of Equations (205), enables the factorization of the map

()

Besides, provided that the radius b > 0 is chosen in the vicinity of the origin, we can find a constant

with

()

for all

, all

close to 0. At last, the factorization of Equation (207) and the lower bounds of Equation (208) give rise to Equation (201).

In the ongoing proposition, we check that the map introduced in Equation (75) represents a shrinking map on some appropriately selected ball in the Banach space examined in Definition 7.

Proposition 7. We fix the sectorial domain and the radius ρ, b as in Lemma 10. Let β, μ > 0 be real numbers fixed as in Section 2.2. Then, assuming that the constants presented in Equation (36) are small enough, for , for all radius ϖ_E > 0 chosen large enough, the map given by Equation (75) is favored with the next two features

(1)
The inclusion
()
is granted, where denotes the closed ball of radius ϖ_E centered at 0 in the space E_{(b, ρ, β, μ)}.
(2)
The 1/2−Lipschitz condition
()
holds for all .

In particular, since the radius ϖ_E can be taken arbitrarily large, we observe that the map turns out to be well defined on the whole space E_{(b, ρ, β, μ)} where the shrinking property (210) holds true.

Proof. Let us focus on the first item of the proposition. We first provide bounds for the forcing term of disclosed in the next

Lemma 11. There exists a constant such that

()

Proof. Owing to the lower bounds of Equation (201) and the definition (38) of together with the bounds of Equation (111), we arrive at

()

for all τ₁ ∈ D_b, τ₂ ∈ D_ρ, all

, where

()

paying regard to the fact that

. At last, the expected bounds of Equation (211) follow from Equation (212) and Definition 7.

In the next lemma, we come up with bounds for the linear part of the map .

Lemma 12. One can find a constant C₂ > 0 such that

()

for all ω(τ₁, τ₂, m) ∈ E_{(b, ρ, β, μ)}.

Proof. Let us take ω ∈ E_{(b, ρ, β, μ)}. We provide bounds for the function

()

By definition of the space E_{(b, ρ, β, μ)}, we notice that

()

for all τ₁ ∈ D_b, all τ₂ ∈ D_ρ and all

. Owing to the assumption (30), we notice that

provided that τ₁ ∈ D_b. Hence,

()

whenever τ₁ ∈ D_b, τ₂ ∈ D_ρ and

. Then, according to the lower bounds of Equation (201) together with Equation (217), we deduce that

()

and bearing in mind the estimates (88), where the map

is introduced in Equation (82), we reach

()

for all τ₁ ∈ D_b, τ₂ ∈ D_ρ and

. At last, we arrive at some constant C₂ > 0 for which the norm bounds

()

holds.

Now, we select the constants

, for

, small enough and take a radius ϖ_E > 0 large enough in a way that the next inequality

()

holds where the constant C₂ > 0 appears in Lemma 12 and

shows up in Lemma 11. Eventually, the bounds of Equation (211) along with Equation (214) under the restriction of Equation (221) trigger the expected inclusion of Equation (209).

In the second part of the proof, we address the shrinking property of Equation (210). Let us choose two arbitrary elements ω₁, ω₂ in the closed ball

whose radius has been prescribed in the first item (209). Owing to Lemma 12, the following inequality

()

holds for the constant C₂ > 0 stemming from Lemma 12. We prescribe the constants

, for

, small enough allowing the next inequality

()

to hold. The Lipschitz property (210) is a straight consequence of Equation (222) under the requirement (223).

In the end, we suitably select the constants , small enough and a radius ϖ_E > 0 large enough in order that both constraints Equations (221) and (223) are granted at once. This induces the two features Equations (209) and (210) for the map .

The next proposition provides a solution to the convolution q−difference Equation (200) inside the space E_{(b, ρ, β, μ)}.

Proposition 8. We prescribe the sectorial domain together with the radius ρ, b as in Lemma 10. Let β, μ > 0 be real numbers fixed as in Section 2.2. Assume that the constants , , are chosen small enough in a suitable way as in Proposition 7. Then, for all radius ϖ_E > 0 large enough, a unique solution ω_b,ρ to the convolution q−difference Equation (200) can be constructed in the space E_{(b, ρ, β, μ)} under the requirement

()

Proof. Select a radius ϖ_E > 0 as in Proposition 7. The closed ball stands for a complete metric space for the distance . Proposition 7 claims that the map induces a contractive map from the metric space into itself. The classical Banach fixed point theorem allows the map to possess a unique fixed point located inside de ball that we denote ω_b,ρ. As a result, the next identity

()

holds provided τ₁ ∈ D_b, τ₂ ∈ D_ρ, for all

. At last, under the conditions imposed, we observe that the convolution q−difference Equation (200) can exactly be rearranged after a division by the map P_m(τ₁, τ₂) as Equation (225). As a consequence, the unique fixed point ω_b,ρ obtained in

fully solves Equation (200). This yields Proposition 8.

5.2. Link between the Solutions and ω_b,ρ to the Convolution q−Difference Equations (65) and (200)

In order to unveil the analytic relation between the two solutions and ω_b,ρ to the same convolution q−difference equation considered in Sections 3.2 and 5.1, we introduce a new auxiliary Banach space.

Definition 8. Let b, ρ > 0 be given positive real numbers and let be an unbounded sector edged at 0 with bisecting direction . We denote the vector space of all continuous maps (τ₁, τ₂, m) ↦ h(τ₁, τ₂, m) on the product , holomorphic relatively to the couple (τ₁, τ₂) on the domain , for which the norm

()

is a finite quantity. The vector space

equipped with the norm

is a Banach space.

In the next proposition, we claim that the map displayed in Equation (75) is well defined on the space where it boasts a 1/2−Lipschitz feature.

Proposition 9. We prescribe the sectorial domain and the radius b, ρ as in Lemma 10. We set the constants β, μ > 0 as in Section 2.2. We select an unbounded sector as in Lemma 4. Then, assuming that the constants introduced in Equation (36) are close enough to 0, for all , the map declared in Equation (75) is well defined on the whole space and is subjected to the next 1/2−Lipschitz condition

()

for all ω₁, ω₂ belonging to

Proof. The proof of Proposition 9 mirrors in the very details one of Proposition 7 and will not be presented in this work in order to avoid redundancy.

The following proposition establish the awaited analytical connection between and ω_b,ρ.

Proposition 10. Let the sectorial domain and the radius b, ρ be prescribed as in Lemma 10. The constants β, μ > 0 are set as in Section 2.2, and the unbounded sector is chosen as in Lemma 4. Then, provided that the constants given by Equation (36) are taken in the vicinity of the origin for all , the next identity

()

holds for all

, all τ₂ ∈ D_ρ, all

. In particular, for given τ₂ ∈ D_ρ and

, the partial map τ₁ ↦ ω_b,ρ(τ₁, τ₂, m) is the analytic continuation of the partial map

on the full disc D_b.

Proof. According to Proposition 2, we know that the map belongs to the Banach space . According to Definition 8 it follows that the restricted map , for , τ₂ ∈ D_ρ, and belongs to . On the other hand, we know from Proposition 8 that the map ω_b,ρ belongs to the space E_{(b, ρ, β, μ)}. As a result, the restricted map (τ₁, τ₂, m) ↦ ω_b,ρ(τ₁, τ₂, m) on also belongs to . Furthermore, according to Equations (119) and (225), we observe in particular that the next two identities

()

holds as functions provided that

, τ₂ ∈ D_ρ, and

. At last, if one sets

and ω₂ = ω_b,ρ in the inequality of Equation (227), it follows from Equation (229) that

()

It implies that

, from which the expected identity Equation (228) follows.

5.3. Statement of the Second Main Result

In this subsection, we exhibit a fine structure for the asymptotic expansion of Gevrey/q−Gevrey type for the solution u(t, z) to Equation (168), which combines both a logarithmic scale and a power scale. The next statement represents the second deed of our work.

Theorem 2. We consider the function u(t, z) displayed in Equation (169), which solves our main initial value problem (168) for vanishing initial data u(0, z) ≡ 0 built up in Theorem 1. Then, the map u(t, z) can be broken up as a sum of two functions

()

where

(1)
the map u₁(t, z) is bounded holomorphic on the domain and possesses a generalized asymptotic expansion of so-called q−Gevrey type in a power scale as t tends to 0. It means that one can distinguish a formal power series
()
with bounded coefficients b_n(t, z) on the domain , which represents a generalized asymptotic expansion of q−Gevrey order k₁ in the scale of monomials of the map u₁(t, z) with respect to t on the domain . Namely, two constants B₁, B₂ > 0 can be singled out for which the next error bounds
()
hold for all integers N ≥ 0, all , provided that .
(2)
The map u₂(t, z) is bounded holomorphic on the domain and carries the null formal series as asymptotic expansion of Gevrey order 1 in a logarithmic scale as t tends to 0. In other words, two constants B₃, B₄ > 0 can be identified in order that the following error bounds
()
hold for all integers N ≥ 0, all , as long as .

Proof. Our idea consists of the splitting of the triple integral representation of u(t, z) given by Equation (169) into three specific contributions

()

where

()

and

()

in a row with

()

where the integration paths are stated as follows

()

along with

()

where the positive real numbers b, ρ > 0 are prescribed in Lemma 10.

In the next first main proposition, we provide asymptotic expansions for the first piece v₁(t, z) relatively to t.

Proposition 11. There exists a sequence of maps g_k(t, z), k ≥ 0, that are well defined and bounded holomorphic relatively to (t, z) on the product which are submitted to the bounds

()

for some well-selected constants M₁, ϖ_E > 0 and R > 0, where

and Δ₁ > 0 are the two constants arising in Equation (18), for all integers k ≥ 0, provided that

and

. For any given natural number N ≥ 0, the next decomposition

()

holds for (t, z) on

, where the remainder term v_1,N+1(t, z) stands for a bounded holomorphic function on

and is monitored by means of the bounds

()

for the constants

appearing in Equation (241) and for a suitable small radius

, as long as

and

Proof. Let b, ρ > 0 be fixed as in Lemma 10. Owing to Proposition 8, we know in particular that the partial map τ₁ ↦ ω_b,ρ(τ₁, τ₂, m) is bounded and analytic on the disc D_b for any prescribed τ₂ ∈ D_ρ and . As a result, we can apply the Taylor formula with integral remainder of some fixed order N ≥ 0 to that function and get the next expansion

()

provided that τ₁ ∈ D_b, τ₂ ∈ D_ρ and

. According to Proposition 10, we know that the function

coincides with the map ω_b,ρ(τ₁, τ₂, m) for

, all τ₂ ∈ D_ρ and

. Hence, from the identity of Equation (244), we deduce the next development

()

for all

, all τ₂ ∈ L_π,ρ/2 and all

. This last Equation (245) enable the expansion of the map v₁(t, z) in the form

()

where

()

for 0 ≤ k ≤ N.

In the next step, we provide upper bounds for the maps g_k(t, z), 0 ≤ k ≤ N. We first need to remind the reader of the next equation

()

for all

which has been applied in our recent work [10], see Lemma 3 therein, from which we deduce the splitting

()

for all

. As a result, one can further break up the term g_k as follows:

()

where

()

In the next lemma, we focus on bounds for the function a_k(t, z).

Lemma 13. For all 0 ≤ k ≤ N, the map a_k(t, z) is well defined and bounded holomorphic with respect to (t, z) on . Furthermore, there exists two constants M₁ > 0 and 0 < R < b such that

()

for all (t, z) on

, provided that 0 < β^′ < β.

Proof. We remind from Proposition 8 that the map ω_b,ρ belongs to the space E_{(b, ρ, β, μ)} and that a constant ϖ_E > 0 can be pinpointed with the bounds

()

provided that τ₁ ∈ D_b, τ₂ ∈ D_ρ and

. Besides, from the classical Cauchy’s equation, we know the next integral representation

()

to hold for τ₂ ∈ D_ρ and

, where the integration is realized along any positively oriented circle C_R centered at 0 with radius R subjected to 0 < R < b. On account of Equation (254) and the bounds of Equation (253), we reach the estimates

()

for all τ₂ ∈ D_ρ, all

. As a result of Equation (255), we arrive at

()

where

()

for all

, all

with 0 < β^′ < β.

In the next lemma, bounds for the second piece of Equation (250) are determined.

Lemma 14. For all 0 ≤ k ≤ N, the map

()

is well defined and stand for a bounded holomorphic function relatively to (t, z) on

. In addition, the next upper bounds

()

hold for all

, where the constants M₁ > 0 and R > 0 are prescribed in Lemma 13 and where

and Δ₁ > 0 are the two constants appearing in Equation (18).

Proof. The technical estimates displayed in the next lemma are crucial.

Lemma 15. The next inequality

()

holds for all

, all integers k ≥ 0, where

and Δ₁ > 0 are the two constants appearing in Equation (18).

Proof. Owing to Equation (18), we first observe that

()

for all

and

. Based on Equation (261), we deduce that

()

where the quantity I_k,|t|,b is derived by performing the change of variable s₁ = r₁/|t| in the integral along the segment [b/2, +∞) above and stands for

()

In the next step, we reach upper bounds for I_k,|t|,b. By coarse upper estimates, we first get

()

Then, at last, we show that the constant I_k,0 can be computed in an exact manner. Indeed, we make the change of variable

()

in the integral I_k,0, which gives rise to

()

On the other hand, we recall the Gaussian identity

()

which is valid for any given real number

and stems from the book [14], Chapter 10, p. 498. This last identity enables the straight computation of Equation (266) as follows:

()

for any integer k ≥ 0.

Eventually, we gather all the above bounds of Equations (262), (264), and (268) and arrive at Equation (260).

With the help of the above lemma, we achieve bounds for the map g_k,1(t, z). Indeed, from Equation (260) in a row with Equation (255), we get

()

for all

for the constant M₁ > 0 defined in Equation (257).

In the next lemma, we address bounds for the remainder part of the expansion (246) for v₁(t, z).

Lemma 16. Let us denote

()

the tail piece of Equation (246). The map v_1,N+1(t, z) is well defined and represents a bounded holomorphic function on the domain

. Moreover, the next estimates

()

hold whenever

and

, for the constant M₁ > 0 defined in Equation (257) and where

is some fixed small radius.

Proof. We first need to upper bound the next quantity

()

relatively to t and N. Indeed, owing to Equation (261), we deduce

()

where the element

is obtained by applying the change of variable s₁ = r₁/|t| in the integral along the segment [0, b/2] overhead and stands for

()

In the next step, we merely observe that

()

where I_N+1,0 is given in the inequality of Equation (264). According to the computation made in Equation (268), we notice that

()

At last, with the combination of Equations (273), (274), (275), and (276), we arrive at

()

provided that

, for any given integer N ≥ 0.

Besides, owing to the classical Cauchy’s equation, the next integral representation

()

holds for all τ₁ ∈ D_b/2, τ₂ ∈ D_ρ, u ∈ [0, 1] and

, where the integration is performed along a positively oriented circle

centered at uτ₁ with small radius

chosen in a way that

. From Equation (278) together with Equation (253), we deduce the useful bounds

()

for all τ₁ ∈ D_b/2, τ₂ ∈ D_ρ, u ∈ [0, 1] and

. Eventually, the gathering of Equations (277) and (279) gives rise to

()

for all

and

where the constant M₁ > 0 is defined in Equation (257).

In conclusion, Proposition 11 ensues from the decompositions Equations (246) and (250) and the collection of Lemmas 13, 14, and 16.

In the second main proposition, we show that the second piece v₂(t, z) has the null formal series as Gevrey asymptotic expansion of order 1 in a logarithmic scale with respect to t.

Proposition 12. The map v₂(t, z) is well defined and bounded holomorphic relatively to (t, z) on the product . Furthermore, for some well-chosen constants and any given integer N ≥ 0, the next error bounds

()

hold provided that

and

, where

and Δ₁ > 0 are the two constants stemming from Equation (18).

Proof. According to Equation (118) in Proposition 2, one can find a constant M_ϖ,b > 0 for which the map is subjected to the next upper bounds

()

provided that

, τ₂ ∈ L_π,ρ/2,∞ and

. Besides, the bounds of Equation (277) for the quantity of Equation (272) in the special case N = 0 yields the next estimates

()

for all

. Furthermore, a constant

can be singled out with

()

as long as

, for R₁ > 0 chosen small enough. In the next step, we remind the reader of the following technical estimates that are taken from Lemma 14 of [10]. Namely, for any given real number M > 0, one can select a constant Q₁ > 0 such that

()

for all integers N ≥ 1, all real numbers r > 0. Based on Equation (285) for the constant

and specific value r = −1/log|t|, we deduce from Equation (284) that

()

for all integers N ≥ 1, provided that

At last, the collection of the bounds of Equations (282), (283), and (286) triggers the next error bounds for the piece v₂(t, z). Namely,

()

for all integers N ≥ 0, whenever

and

In the last principal proposition, the third piece v₃(t, z) is shown to have the null formal series as the asymptotic expansion of q−Gevrey order k₁ in the scale of monomials relatively to t.

Proposition 13. The map v₃(t, z) is well defined and bounded holomorphic relatively to (t, z) on the product . In addition, for some suitable constants ϖ > 0, M_δ,1,b, M_δ,2,b > 0, M₃ > 0, and any given integer N ≥ 0, the next error bounds

()

hold provided that

and

, where

and Δ₁ > 0 are the two constants appearing in Equation (18).

Proof. We further break up the integral v₃(t, z) in two parts

()

where

()

with

and for

()

along the segment

As stated in Equation (118) in Proposition 2, the map

is subjected to the next upper bounds

()

provided that

, and

. On the other hand, we need the next technical upper bounds.

Lemma 17. One can single out two constants M_δ,1,b, M_δ,2,b > 0 such that

()

provided that

and

Proof. For and , we first expand

()

along with

()

and

()

Since log(1 + x) ~ x holds as x tends to 0 and bearing in mind the classical growth comparison

, we get from Equations (295) and (296) two constants M_δ,1,b, M_δ,2,b > 0 with

()

for all |τ₁| ≥ b/2. Eventually, the gathering of Equations (261) and (297) with the expansion of Equation (294) yields the awaited estimates of Equation (293).

In the next lemma, we exhibit q−Gevrey type estimates on both segments and .

Lemma 18. The next two q−Gevrey type estimates hold.

(1)
On the segment , we get that
()
holds for all , all , for all integers N ≥ 0.
(2)
On the segment , we arrive at
()
provided that , all , for all integers N ≥ 0.

Proof.

(1)
Consider and . In particular, we notice that b/2 ≤ |τ₁| ≤ 1 and |t| < R₁ < 1. It follows that

()

As a result, the inequality of Equation (293) becomes

()

(2)
Let us take . In particular |τ₁| ≥ 1. We select R₁ > 0 small enough and fulfilling (122) in a way that

()

for all

. The inequality of Equation (293) is then changed into

()

The next estimates have been presented in Lemma 12 of our recent work [10]. Namely, for any prescribed real number , the next inequality

()

occurs for all integers N ≥ 1, all positive real numbers x > 0. In particular, the next upper bounds

()

along with

()

hold for all

, for all integers N ≥ 0.

At last, the q−Gevrey type bounds of Equation (298) result from Equation (301) together with Equation (305) and the combination of Equation (303) with Equation (306) yields of Equation (299).

In the last part of the proof, we can now provide upper bounds for each piece v_3.1(t, z) and v_3.2(t, z). Namely, based on Equations (292), (298), and (299), we get

()

along with

()

for the constant

()

for all integers N ≥ 0, whenever

and

Eventually, the splitting (289) together with the above upper estimates of Equations (307) and (308) promotes the expected bounds of Equation (288).

We return to the proof of Theorem 2. On the ground of the decomposition of Equation (235), we set

()

According to Propositions 11 and 13, we observe that u₁(t, z) represents a bounded holomorphic map on the domain

. Moreover, u₁ is submitted to error bounds of the form Equation (233) for the sequence of functions b_n(t, z), n ≥ 0 given by b_n(t, z) = g_n(t, z)/tⁿ, which represent bounded holomorphic maps on the domain

, owing to the upper bounds of Equation (241).

On the other hand, we assign

()

As claimed by Proposition 12, we check that u₂(t, z) stands for a bounded holomorphic function on

. Furthermore, u₂ is subjected to error bounds shaped in Equation (281). Theorem 2 is established.

6. Conclusion

In this work, we have considered a linear initial value q−difference differential problem, which possesses an irregular singularity in complex time at the origin. This problem is a q−analog of the recent study [1] where the author has constructed logarithmic type solutions to singularly perturbed partial differential equations by means of double complete and truncated Laplace transforms and has built asymptotic expansions in both power and logarithmic scales in the perturbation parameter and time.

In the present investigation, logarithmic type solutions are achieved by means of double q−Laplace and Laplace transforms, both complete in the variables t and 1/log(t). This is one of the striking novelties compared to [1]. Asymptotic expansions of Gevrey type in the logarithmic scale are obtained in a similar manner as in [1] through the classical version of the Ramis-Sibuya theorem. However, such an approach breaks down for the construction of asymptotic expansions in the power scale . This is the second compelling novelty in contrast to [1]. For that purpose, another walk-through has been favored, and new kinds of asymptotic expansions have been unveiled with mixed-type remainders involving q−Gevrey growth in the power scale and Gevrey of order 1 rate in the logarithmic scale.

Disclosure

The present work is registered as a preprint on preprints.org where it is quoted as [15].

Conflicts of Interest

The author declares that there is no conflicts of interest.

Acknowledgments

The work of the author is supported by the University of Lille, France. Open Access funding enabled and organized by COUPERIN CY23.

Open Research

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value Problem

Abstract

1. Introduction

2. Setup of the Main Initial Value Problem and an Associated Set of q−Difference-Differential Problems with Homography Action

2.1. Accounts on q−Laplace Transforms of Order k and Fourier Inverse Maps

2.2. Layout of the Main Problem

2.3. A Set of Related q−Difference-Differential Equations with an Homography Action

3. Analytic Solutions to the Associated Set of q−Difference and Differential Problems under Homography Action

3.1. Profile of the Analytic Solutions and Joint Convolution q−Difference Equations

3.2. Solving the Convolution q−Difference Equation (65) on Unbounded Sectors and Half Strips

3.3. Analytic Solutions to the Auxiliary Equations (54) and (56)

4. Construction of a Holomorphic Solution to the Main Initial Value Problem (21) and Its Gevrey Asymptotic Expansion Relatively to Complex Time t in Logarithmic Scale

4.1. A Finite Set of Genuine Solutions to Related Initial Value Problems

4.2. Gevrey Asymptotic Expansions for the Bounded Holomorphic Solutions to the Family of Auxiliary Problems (139) and (141)

4.3. Statement of the First Main Result

4.4. Computational Features Related to the Formal Power Series (162)

5. Fine Structure of Gevrey/q−Gevrey Asymptotic Expansions in Combined Power and Logarithmic Scales for the Holomorphic Solution to the Initial Value Problem (29)

5.1. Solving the Convolution q−Difference Equation (65) on Some Neighborhood of the Origin

5.2. Link between the Solutions and ω_b,ρ to the Convolution q−Difference Equations (65) and (200)

5.3. Statement of the Second Main Result

6. Conclusion

Disclosure

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

Information

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Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value Problem

Abstract

1. Introduction

2. Setup of the Main Initial Value Problem and an Associated Set of q−Difference-Differential Problems with Homography Action

2.1. Accounts on q−Laplace Transforms of Order k and Fourier Inverse Maps

2.2. Layout of the Main Problem

2.3. A Set of Related q−Difference-Differential Equations with an Homography Action

3. Analytic Solutions to the Associated Set of q−Difference and Differential Problems under Homography Action

3.1. Profile of the Analytic Solutions and Joint Convolution q−Difference Equations

3.2. Solving the Convolution q−Difference Equation (65) on Unbounded Sectors and Half Strips

3.3. Analytic Solutions to the Auxiliary Equations (54) and (56)

4. Construction of a Holomorphic Solution to the Main Initial Value Problem (21) and Its Gevrey Asymptotic Expansion Relatively to Complex Time t in Logarithmic Scale

4.1. A Finite Set of Genuine Solutions to Related Initial Value Problems

4.2. Gevrey Asymptotic Expansions for the Bounded Holomorphic Solutions to the Family of Auxiliary Problems (139) and (141)

4.3. Statement of the First Main Result

4.4. Computational Features Related to the Formal Power Series (162)

5. Fine Structure of Gevrey/q−Gevrey Asymptotic Expansions in Combined Power and Logarithmic Scales for the Holomorphic Solution to the Initial Value Problem (29)

5.1. Solving the Convolution q−Difference Equation (65) on Some Neighborhood of the Origin

5.2. Link between the Solutions and ωb,ρ to the Convolution q−Difference Equations (65) and (200)

5.3. Statement of the Second Main Result

6. Conclusion

Disclosure

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

Related

Information

5.2. Link between the Solutions and ω_b,ρ to the Convolution q−Difference Equations (65) and (200)